There is a pleasant touch of mystery in the following method of
discovering a person’s age:--Ask any such subjects of your curiosity to
write down the tens digit of the year of their birth, to multiply this
by 5, to add 2 to the product, to multiply this result by 2, and finally
to add the units digit of their birth year. Then, taking the paper from
them, subtract the sum from 100. This will give you their age in 1896,
from which their present age is easily determined.
No. XXXIX.--CHESS CAMEO
BY FRANK HEALEY
_The “Bristol Prize Problem”_
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| n |...| |...| N |.p.| |
+---+---+---+---+---+---+---+---+
| |.N.| |...| |...| Q |...|
+---+---+---+---+---+---+---+---+
|...| b |.k.| P |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| p |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|.P.| |.P.| |...| R |...| |
+---+---+---+---+---+---+---+---+
| |...| |.P.| |...| P |.K.|
+---+---+---+---+---+---+---+---+
|.B.| |...| R |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
FUR AND FEATHERS
As I came in after a day among the birds and rabbits, the keeper asked
me--“Well, sir, what sport?” I replied, “36 heads and 100 feet.” It took
him some time to calculate that I had accounted for 22 birds and 14
rabbits.
No. XL.--CHESS CAMEO
BY J. E. CAMPBELL
_Splendid Strategy_
BLACK
+---+---+---+---+---+---+---+---+
| |...| |.N.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| p |.p.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| |.k.| p |.P.| |.n.| Q |
+---+---+---+---+---+---+---+---+
| |...| p |.N.| |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| B |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| K |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
JUGGLING WITH THE DIGITS
The nine digits can be arranged to form fractions equivalent to
1 1 1 1 1 1 1
- - - - - - -
3 4 5 6 7 8 9
thus:--
5823 1 7956 1 2973 1 2943 1 5274 1
----- = - ----- = - ----- = - ----- = - ----- = -
17469 3 31824 4 14865 5 17658 6 36918 7
9321 1 8361 1
----- = - ----- = -
74568 8 75249 9
No. XLI.--CHESS CAMEO
BY W. GRIMSHAW
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.N.| |...| K |...| |...| |
+---+---+---+---+---+---+---+---+
| |.b.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| p |.p.| p |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.P.| P |
+---+---+---+---+---+---+---+---+
| |.Q.| |...| |...| R |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
No. XLII.--CHESS CAMEO
BY S. LOYD
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|...| |.K.| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.p.| |
+---+---+---+---+---+---+---+---+
| |.Q.| |...| |.N.| k |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
A MOTOR PROBLEM
This motor problem will be new and amusing to many readers:--
Let _m_ be the driver of a motor-car, working with velocity _v_. If a
sufficiently high value is given to _v_, it will ultimately reach _pc_.
In most cases _v_ will then = _o_. For low values of _v_, _pc_ may be
neglected; but if _v_ be large it will generally be necessary to square
_pc_, after which _v_ will again assume a positive value.
By a well-known elementary theorem, _pc_ + _lsd_ = (_pc_)², but the
squaring may sometimes be effected by substituting _x_³ (or × × ×) for
_lsd_. This is preferable, if _lsd_ is small with regard to _m_. If
_lsd_ be made sufficiently large, _pc_ will vanish.
Now if _jp_ be substituted for _pc_ (which may happen if the difference
between _m_ and _pc_ be large) the solution of the problem is more
difficult. No value of _lsd_ can be found to effect the squaring of
_jp_, for, as is well-known, (_jp_)² is an impossible quantity.
No. XLIII.--CHESS CAMEO
BY J. G. CAMPBELL
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| p |...| Q |
+---+---+---+---+---+---+---+---+
| |...| |.k.| N |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.N.| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| P |...| B |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.P.| p |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.K.| |.B.|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
No. XLIV.--CHESS CAMEO
BY FRANK HEALEY
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |.K.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.P.| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.Q.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
A NEAT METHOD OF DIVISION
To divide any sum easily by 99, cut off the two right-hand figures of
the dividend and add them to all the others. Set down the result of this
in line below, and then repeat this process until no figures remain on
the left to be thus dealt with.
Now draw a line down between the tens and hundreds columns, and add all
up on the left of it, thus:--
8694 | 32 120 | 78
87 | 26 1 | 98
1 | 13 | 99
| 14 ------------
------------- 121 and 99 over.
8782 and 14 over. In other words, 122.
The last number on the right of the lines shows always the remainder. If
this should appear as 99 (as in the second example above), add one to
the number on the left.
No. XLV.--CHESS CAMEO
BY BLUMENTHAL AND KUND
BLACK
+---+---+---+---+---+---+---+---+
| |...| K |...| |.R.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| B |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.r.| n |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| k |...| p |...| |...| |.Q.|
+---+---+---+---+---+---+---+---+
|...| |...| N |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
No. XLVI.--CHESS CAMEO
BY A. F. MACKENZIE
_A Prize Problem_
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |.R.| |...|
+---+---+---+---+---+---+---+---+
|...| |.p.| |.B.| |...| Q |
+---+---+---+---+---+---+---+---+
| n |...| |.R.| |...| |.b.|
+---+---+---+---+---+---+---+---+
|.p.| |.k.| P |...| |...| |
+---+---+---+---+---+---+---+---+
| K |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
A SMART SCHOOLBOY
The question, “How many times can 19 be subtracted from a million?” was
set by an examiner, who no doubt expected that the answer would be
obtained by dividing a million by 19. One bright youth, however, filled
a neatly-written page with repetitions of
1,000,000 1,000,000 1,000,000
19 19 19
--------- --------- ---------
999,981 999,981 999,981
and added at the foot of the page, “_N.B._--I can do this as often as
you like.”
There was a touch of unintended humour in this, for, after all, the boy
gave a correct answer to a badly worded question.
No. XLVII.--CHESS CAMEO
BY A. CYRIL PEARSON
BLACK
+---+---+---+---+---+---+---+---+
| |.N.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| r |.n.| |...| n |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.R.| |...|
+---+---+---+---+---+---+---+---+
|...| |.P.| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| p |.P.| |...| |...|
+---+---+---+---+---+---+---+---+
|.Q.| |.P.| |.K.| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
No. XLVIII.--CHESS CAMEO
BY FRANK HEALEY
_Quite a Gem_
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |.b.|
+---+---+---+---+---+---+---+---+
|...| |...| n |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.K.| |...| |...| |.Q.| |
+---+---+---+---+---+---+---+---+
| |.P.| |.k.| b |...| P |...|
+---+---+---+---+---+---+---+---+
|.R.| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| B |...| |...| |.P.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
LEWIS CARROLL’S SHORT CUT
Here is a very smart and very simple method of dividing any multiple of
9 by 9, from the fertile brain of Lewis Carroll:--Place a cypher over
the final figure, subtract the final figure from this, place the result
above in the tens place, subtract the original tens figure from this,
and so on to the end. Then the top line, excluding the intruded cypher,
gives the result desired. Thus:--
36459 ÷ 9 = 4051,0
= 4051.
36459
No. XLIX.--CHESS CAMEO
BY A. BAYERSDORFER
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| K |...| |
+---+---+---+---+---+---+---+---+
| |...| |.k.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.p.| B |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| Q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
ANOTHER FREAK OF FIGURES
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
123456 × 8 + 6 = 987654
1234567 × 8 + 7 = 9876543
12345678 × 8 + 8 = 98765432
123456789 × 8 + 9 = 987654321
No. L.--CHESS CAMEO
BY J. DOBRUSKY
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |...| B |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| q |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| k |...| |...| |
+---+---+---+---+---+---+---+---+
| |.R.| N |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| K |.R.| |...| |...| |
+---+---+---+---+---+---+---+---+
| Q |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.n.| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
DIVINATION BY NUMBERS
Here is one of the methods by which we can readily discover a number
that is thought of. The thought-reader gives these directions to his
subject: “Add 1 to three times the number you have thought of; multiply
the sum by 3; add to this the number thought of; subtract 3, and tell me
the remainder.” This is always ten times the number thought of. Thus, if
6 is thought of--6 × 3 + 1 = 19; 19 × 3 = 57; 57 + 6 - 3 = 60, and 60 ÷
10 = 6.
No. LI.--CHESS CAMEO
BY KONRAD BAYER
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| p |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.k.| |.K.| |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.p.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.R.| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
COINCIDENCES
Here is a curious rough rule for remembering distances and sizes:--
The diameter of the earth multiplied by 108 gives approximately the
sun’s diameter. The diameter of the sun multiplied by 108 gives the mean
distance of the earth from the sun. The diameter of the moon multiplied
by 108 gives the mean distance of the moon from the earth.
No. LII.--CHESS CAMEO
BY J. BERGER
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| B |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| p |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| k |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| Q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.p.| |.K.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
PERSONAL ARITHMETIC
Says Giles, “My wife and I are two,
Yet faith I know not why, sir.”
Quoth Jack, “You’re ten, if I speak true,
She’s one, and you’re a cypher!”
No. LIII.--CHESS CAMEO
BY H. F. L. MEYER
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| R |...| |...|
+---+---+---+---+---+---+---+---+
|...| k |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| b |.p.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| p |...| P |...| |...| |
+---+---+---+---+---+---+---+---+
| |.K.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.B.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| Q |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
DIVISION BY SUBTRACTION
Here is a curious and quite uncommon method of dividing any multiple of
11 by 11.
Set down the multiple of 11, place a cypher under its last figure, draw
a line, and subtract, placing the first remainder under the tens place.
Subtract this from the next number in order, and so on throughout,
adding in always any number that is carried. Thus:--
363 56408 375034
0 0 0
--- ----- ------
33 5128 34094
No. LIV.--CHESS CAMEO
BY FRANK HEALEY
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.B.| |...| |...| |...| p |
+---+---+---+---+---+---+---+---+
| |.r.| p |.p.| |...| b |...|
+---+---+---+---+---+---+---+---+
|...| |.k.| n |...| |.Q.| |
+---+---+---+---+---+---+---+---+
| P |...| |...| p |...| |...|
+---+---+---+---+---+---+---+---+
|.K.| |...| |.P.| |...| N |
+---+---+---+---+---+---+---+---+
| B |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
LUCK IN ODD NUMBERS
Perhaps the old saying, “there is luck in odd numbers,” may have some
connection with the curious fact that the sum of any quantity of
consecutive odd numbers, beginning always with 1, is the square of that
number. Thus:--
1 + 3 + 5 = 9 = 3 × 3.
1 + 3 + 5, etc., up to 17 = 81 = 9 × 9.
1 + 3 + 5, etc., up to 99 = 2500 = 50 × 50.
No. LV.--CHESS CAMEO
BY FRANK HEALEY
BLACK
+---+---+---+---+---+---+---+---+
| |.K.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|.n.| |...| |.p.| |...| |
+---+---+---+---+---+---+---+---+
| |.N.| |.k.| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |.N.| |...| |...| |
+---+---+---+---+---+---+---+---+
| |.P.| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |.n.| Q |.P.| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
THE VERSATILE NUMBER
In the number 142857, if the digits which belong to it are in succession
transposed from the first place to the end, the result is in each case a
multiple of the original number. Thus:--
285714 = 142857 × 2
428571 = 142857 × 3
571428 = 142857 × 4
714285 = 142857 × 5
857142 = 142857 × 6
No. LVI.--CHESS CAMEO
BY J. E. CAMPBELL
BLACK
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| p |...| p |...| B |
+---+---+---+---+---+---+---+---+
| |...| |...| |.k.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| K |...| N |...| |
+---+---+---+---+---+---+---+---+
| |...| |...| |...| |.P.|
+---+---+---+---+---+---+---+---+
|...| |...| |...| |.R.| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in three moves.
A PARADOX
By the following simple method, a plausible attempt is made to prove
that 1 is equal to 2:--
Suppose that _a_ = _b_, then
_ab_ = _a_²
∴ _ab_ - _b_² = _a_² - _b_²
∴ _b_(_a_ - _b_) = (_a_ + _b_)(_a_ - _b_)
∴ _b_ = _a_ + _b_
∴ _b_ = 2_b_
∴ 1 = 2
This process only proves in reality that 0 × 1 = 0 × 2, which is true.
No. LVII.--CHESS CAMEO
_Double First Prize_
BY A. CYRIL PEARSON
BLACK
+---+---+---+---+---+---+---+---+
| |...| b |...| |...| |...|
+---+---+---+---+---+---+---+---+
|...| |...| p |.K.| |...| p |
+---+---+---+---+---+---+---+---+
| |...| p |...| |.p.| |...|
+---+---+---+---+---+---+---+---+
|...| |...| B |.k.| |...| |
+---+---+---+---+---+---+---+---+
| |.b.| |...| |...| P |...|
+---+---+---+---+---+---+---+---+
|.N.| |.N.| |.P.| |...| |
+---+---+---+---+---+---+---+---+
| |.R.| |.P.| |.R.| |...|
+---+---+---+---+---+---+---+---+
|.b.| q |...| |...| |...| |
+---+---+---+---+---+---+---+---+
WHITE
White to play, and mate in four moves.
QUICK CALCULATION
Few people know a very singular but simple method of calculating rapidly
how much any given number of pence a day amounts to in a year. The rule
is this:--Set down the given number of pence as pounds; under this place
its half, and under that the result of the number of original pence
multiplied always by five. Take, for example, 7d a day:--
£7 0 0
3 10 0
2 11
---------
£10 12 11
---------
The reason for this is evident as soon as we remember that the 365 days
of a year may be split up into 240, 120, and 5, and that 240 happens to
be the number of pence in a pound.
SCIENCE AT PLAY
No. LVIII.--THE GEARED WHEELS
A small wheel with ten teeth is geared into a large fixed wheel which
has forty teeth. This small wheel, with an arrow mark on its highest
cog, is revolved completely round the large wheel. How often during its
course is the arrow pointing directly upwards? Here is a diagram of the
starting position.
[Illustration]
No. LIX.--ADVANCING BACKWARDS
Here is a most curious and interesting question:--When an engine is
drawing a train at full speed from York to London, what part of the
train at any given moment is moving _towards York_?
At any time, when the engine is drawing a train at full speed from York
to London, that part of the flange of each wheel which is for the moment
at its lowest is actually _moving backwards towards York_.
[Illustration]
For any point, such as A, on the circumference of the tyre, describes in
running along a series of curves, as shown by _full_ lines in the
diagram; and any point, B, on the outer edge of the flange, follows a
path shown by the _dotted_ curves.
If these lines are followed round with a pencil in the direction of the
arrows, it will be found that the point on the flange actually moves
_backwards_ as it passes _below the track_, while the point A, as it
completes each curve, is _at rest_ for the instant on the track, just
before it starts afresh. The speed of the train does not affect these
very curious facts.
No. LX.--THE FIFTEEN BRIDGES
In the subjoined diagram A and B represent two islands, round which a
river runs as is indicated, with fifteen connecting bridges, that lead
from the islands to the river’s banks.
[Illustration]
Can you contrive to pass in turn over all these bridges without ever
passing over the same one twice?
ARRANGING THE DIGITS
In a school where two boys were taught to think out the bearings of
their work, a sharp pupil remarked that 100 is represented on paper by
the smallest digit and two cyphers, which are in themselves symbols of
nothing. The master, quick to catch any signs of mental activity, took
the opportunity to propound to his class the following ingenious
puzzle:--How can the sum of 100 be represented exactly in figures and
signs by making use of all the nine digits in their reverse order? This
is how it is done:--
9 × 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 100.
Another ingenious method of using the nine digits, so that by simple
addition they sum up to exactly 100, and each is used once only, is
this:--
15 + 36 + 47 = 98 + 2 = 100
Here is another arrangement by which the nine digits written in their
inverse order can be made to represent exactly 100:--
98 - 76 + 54 + 3 + 21 = 100.
Here is yet another way of arriving at 100 by using each of the digits,
this time with an 0:--
40¹⁄₂
59³⁸⁄₇₆
--------
100
No. LXI.--LOOPING THE LOOP
Here is quite a pretty scientific experiment, which any one of a handy
turn can construct and arrange:--
[Illustration]
The spiral track is formed of two wires bent, and connected by curved
cross-pieces. The upper twist is turned so that the ball starts on a
horizontal course.
During the accelerated descent the ball acquires momentum enough to keep
it on the vertical track, held outwardly against the wires by
centrifugal force.
Convenient proportions are: height of spiral two feet, diameter six
inches, and wire rails three-quarters of an inch apart.
No. LXII.--A MECHANICAL BIRD
A close approach to an ideal flying machine can be made with a little
ingenuity. Two Y-shaped standards, secured to the backbone rod, support
two wires which carry wings of thin silk, provided with light stays, and
connected at their inner corners with the backbone by threads.
[Illustration]
Rubber bands are attached to a loop on the inner end of the crank shaft,
and secured to a post at the rear. These are twisted by turning the
shaft with the cross wire, and when the tension is released the wings
beat the air and carry the bird forward. It is known as Penaud’s
mechanical bird, and has been sold as an attractive toy.
No. LXIII.--LINE OF SWIFTEST DESCENT
A simple apparatus constructed on the lines of this illustration will
give an interesting proof of the laws which govern falling bodies on an
inclined plane or on a curved path.
[Illustration]
In the case of the inclined plane the ball is governed by the usual law
which controls falling bodies. In that of the concave circular curve, as
it is accelerated rapidly at the start, it makes its longer journey in
quicker time. In the case of the cycloidal curve it acquires a high
velocity. This curve has therefore been called “the curve of swiftest
descent,” as a falling body passes over it in less time than upon any
path except the vertical.
No. LXIV.--A CENTRIFUGAL RAILWAY
Here is another very simple and pretty illustration of the natural
forces which come into play in “looping the loop.”
[Illustration]
This scientific toy on a small scale may be easily made, if care is
taken that the height of the higher end of the rails is to the height of
the circular part in a greater ratio than 5 to 4.
A ball started at the higher end follows the track throughout, and at
one point is held by centrifugal force against the under side of the
rails, against the force of gravity.
No. LXV.--A QUESTION OF GRAVITY
If a ball is fired point blank from a perfectly horizontal gun, and
travels half a mile over a level plain before it touches ground, and
another similar ball is at the same moment dropped from the same height
by some mechanical means, the two balls will touch ground
simultaneously. The flight, however long, of one through the air has no
influence upon the force of gravity, which draws it earthward at the
same resistless rate as it draws the other that is merely dropped.
[Illustration]
A SHORT CUT
A quick method of multiplying any number of figures by 5 is to divide
them by 2, annexing a cypher to the result when there is no remainder,
and if there is any remainder annexing a 5. Thus:--
464 × 5 = 2320; 464 ÷ 2 = 232, annex 0, = 2320.
753 × 5 = 3765; 753 ÷ 2 = 376, annex 5, = 3765.
No. LXVI.--A DUCK HUNT
A duck begins to swim round the edge of a circular pond, and at the same
moment a water spaniel starts from the middle of the pond in pursuit of
it.
[Illustration]
If both swim at the same pace, how must the dog steer his course so that
he is sure in any case to overtake the duck speedily?
THE MAGIC OF DATES
LOUIS NAPOLEON, EMPEROR 1852
1852 1852 1852
date 1 date 1 date 1
of 8 of 8 of 8
his 0 Empress’s 2 their 5
birth 8 birth 6 marriage 3
---- ---- ----
1869 1869 1869
Thus, by a most remarkable series of coincidences, the principal dates
of the Emperor and Empress of the French added, as is shown above, to
the year of the Emperor’s accession, express in each instance the year
before his fall.
No. LXVII.--GEOMETRY WITH DOMINOES
In this domino diagram we have a pretty and practical proof that the
squares of the sides containing the right angle in any right-angled
triangle are together equal to the square of the side opposite to the
right angle.
[Illustration]
Each stone forms two squares, and it is easily seen that the number of
squares which make up the whole square on the line opposite the right
angle are equal to the number of those which make up the two whole
squares on the lines which contain that angle.
A second point to be noticed is that the number of pips on the large
square are equal to the number on the other two squares combined, an
arrangement of the stones which forms quite a game of patience to
reproduce, if this pattern is not at hand.
No. LXVIII.--TO COLOUR MAPS
Four colours at most are needed to distinguish the surfaces of separate
districts on any plane map, so that no two with a common boundary are
tinted alike.
[Illustration]
On this diagram A, B, and C, are adjoining districts, on a plane
surface, and X borders, in one way or another, upon each.
It is clearly impossible to introduce a fifth area which shall so adjoin
these four districts as to need another tint.
A FREAK OF FIGURES
Here is another freak of figures:--
9 × 1 - 1 = 8
9 × 21 - 1 = 188
9 × 321 - 1 = 2888
9 × 4321 - 1 = 38888
9 × 54321 - 1 = 488888
9 × 654321 - 1 = 5888888
9 × 7654321 - 1 = 68888888
9 × 87654321 - 1 = 788888888
9 × 987654321 - 1 = 8888888888
No. LXIX.--THE TETHERED BIRD
A bird made fast to a pole six inches in diameter by a cord fifty feet
long, in its flight first uncoils the cord, _keeping it always taut_,
and then recoils it in the reverse direction, rewinding the coils close
together. If it starts with the cord fully coiled, and continues its
flight until it brings up against the pole, how far does it fly in its
double course?
[Illustration]
STRIKE IT OUT
Ask a person to write down in a line any number of figures, then to add
them all together as units, and to subtract the result from the sum set
down. Let him then strike out any one figure, and add the others
together as units, telling you the result.
If this has been correctly done, the figure struck out can always be
determined by deducting the final total from the multiple of 9 next
above it. If the total happens to be a multiple of 9, then a 9 was
struck out.
No. LXX.--THE MOVING DISC AND THE FLY
A fly, starting from the point A, just outside a revolving disc, and
always making straight for its mate at the point B, crosses the disc in
four minutes, while the disc is revolving twice. What effect has the
revolution of the disc on the path of the fly?
[Illustration]
A MAGIC SQUARE
This Magic Square is so arranged that the product of the continued
multiplication of the numbers in each row, column, or diagonal is 4096,
which is the cube of the central 16.
+---+---+---+
| 8|256| 2|
+---+---+---+
| 4| 16| 64|
+---+---+---+
|128| 1| 32|
+---+---+---+
No. LXXI.--A SHUNTING PUZZLE
The railway, D E F, has two sidings, D B A and F C A, connected at A.
The rails at A, common to both, are long enough to hold a single wagon
such as P or Q, but too short to admit the whole of the engine R, which,
if it runs up either siding, must return the same way.
[Illustration]
How can the engine R be used to interchange the wagons P and Q without
allowing any flying shunts?--From _Ball’s Mathematical Recreations_.
No. LXXII.--A CURIOUS FACT
It is a little known and very interesting fact that an equilateral
triangle can easily be drawn by rule of thumb in the following
way:--Take a triangle of any shape or size, and on each of its sides
erect an equilateral triangle. Find and join the centres of these, and a
fourth equilateral triangle is always thus formed, as shown by the
dotted lines.
[Illustration]
These centres are _centres of gravity_, and they are symmetrically
distributed around the centre of gravity of the original triangle.
The figure formed by joining them must therefore be symmetrical, and, as
in this case, it is a triangle, it _must be_ always equilateral.
No. LXXIII.--TRY THIS EXPERIMENT
There can be no better instance of how the eye may be deceived than is
so strikingly afforded in these very curious diagrams:--
[Illustration]
The square which obviously contains sixty-four small squares, is to be
cut into four parts, as is shown by the thicker lines. When these four
pieces are quite simply put together, as shown in the second figure,
there seem to be sixty-five squares instead of sixty-four.
This phenomena is due to the fact that the edges of the four pieces,
which lie along the diagonal A B, do not exactly coincide in direction.
In reality they _include a very narrow diamond_, not easily detected,
whose area is just equal to that of one of the sixty-four small squares.
FIGURES IN SWARMS
Very curious are the results when the nine digits in reverse order are
multiplied by 9 and its multiples up to 81. Thus:--
987654321 × 9 = 8888888889
× 18 = 17777777778
× 27 = 26666666997
× 36 = 35555555556
× 45 = 44444444445
× 54 = 53333333334
× 63 = 62222222223
× 72 = 71111111112
× 81 = 80000000001
It will be seen that the figures by which the reversed digits are
multiplied reappear at the beginning and end of each result except the
first, and that the figures repeated between them are to be found by
dividing the divisors by 9 and subtracting the result from 9. Thus, 54 ÷
9 = 6, and 9 - 6 = 3.
No. LXXIV.--A TRIANGLE OF TRIANGLES
[Illustration]
In this nest of triangles there are no less than six hundred and
fifty-three distinct triangles of various shapes and sizes.
No. LXXV.--PHARAOH’S SEAL
In a chamber of the Great Pyramid an ancient Egyptian jar was found,
marked with the device now known as Pharaoh’s seal.
[Illustration]
Can you count the number of triangles or pyramids, of many sizes, but
all of similar shape that are expressed on it? Solvers should draw the
figure on a larger scale.
MAKING CUBES
It is interesting to note that the repeated addition of odd numbers to
one another can be so arranged as to produce cube numbers in due
sequence. Thus:--
1 = 1 × 1 × 1
3 + 5 = 2 × 2 × 2
7 + 9 + 11 = 3 × 3 × 3
13 + 15 + 17 + 19 = 4 × 4 × 4
21 + 23 + 25 + 27 + 29 = 5 × 5 × 5
and so on, to any extent.
No. LXXVI.--ROUND THE GARDEN
In a large old-fashioned garden walks were arranged round a central
fountain in the shape of a Maltese cross.
[Illustration]
If four persons started at noon from the fountain, walking round the
four paths at two, three, four and five miles an hour respectively, at
what time would they meet for the third time at their starting-point, if
the distance on each track was one-third of a mile?
A NICE SHORT CUT
When the tens of two numbers are the same, and their units added
together make ten, multiply the units together, increase one of the tens
by unity, and multiply it by the other ten. The result is the product of
the two original numbers, if the first result follows the other. Thus:--
43 × 47 = 2021.
No. LXXVII.--A JOINER’S PUZZLE
Can you cut Fig. A into two parts, and so rearrange these that they form
either Fig. B or Fig. C?
+--------------+
| |
+--------------+ | +--+
| | | |
| | +-----------------+ | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | | |
| | | | +--+ |
| | | | | |
+--------------+ +-----------------+ +--------------+
A B C
The two parts of A must not be _turned round_ to form B or C, but must
retain their original direction.
A CALCULATION
Coal may fail us, but we can never run short of material for “words that
burn.” It has been calculated that if a man could read 100,000 words in
an hour, and there were 4,650,000 men available, they could not
pronounce the possible variations which could be formed from the
alphabet in 70,000 years!
A PARADOX
It is possible, in a sense, by the following neat method, to take 45
from 45, and find that 45 remains:--
987654321 = 45.
123456789 = 45.
---------
864197532 = 45.
No. LXXVIII.--THE BROKEN OCTAGON
Cut out in stiff cardboard four pieces shaped as Fig. 1, four as Fig. 2,
and four as Fig. 3, taking care that they are all exactly true to
pattern in shape and proportion to one another.
[Illustration]
Now see whether you can put the twelve pieces together so as to form a
perfect octagon.
PROPERTIES OF SEVEN
Here is a proof that 7, if it cannot rival the mystic 9, has quaint
properties of its own:--
15873 × 7 = 111111
31746 × 7 = 222222
47619 × 7 = 333333
63492 × 7 = 444444
79365 × 7 = 555555
95238 × 7 = 666666
111111 × 7 = 777777
126984 × 7 = 888888
142857 × 7 = 999999
A SWARM OF EIGHTS
Here is an arithmetical curiosity:--
9 × 9 + 7 = 88
9 × 98 + 6 = 888
9 × 987 + 5 = 8888
9 × 9876 + 4 = 88888
9 × 98765 + 3 = 888888
9 × 987654 + 2 = 8888888
9 × 9876543 + 1 = 88888888
9 × 98765432 + 0 = 888888888
No. LXXIX.--AT A DUCK POND
A farmer’s wife kept a pure strain of Aylesbury ducks for market on a
square pond, with a duck-house at each corner. As trade grew brisk she
found that she must enlarge her pond. An ingenious neighbour undertook
to arrange this without altering the shape of the pond, and without
disturbing the duck-houses. What was his plan?
○-------------○
| |
| |
| |
| |
| |
| |
○-------------○
STRANGE SUBTRACTION
It would seem impossible to subtract 69 from 55, but it can be arranged
thus, with six as a remainder:--
SIX IX XL
IX X L
-------------
S I X
=============
No. LXXX.--ALL ON THE SQUARE
Cut out in cardboard twenty triangular pieces exactly the size and shape
of this one, and try to place them together so that they form a perfect
square.
[Illustration]
ANOTHER MYSTIC NUMBER
The decimal equivalent of ¹⁄₁₃ is .076923. This (omitting the point),
multiplied by 1, 3, 4, 9, 10, or 12, yields results in which the same
figures appear in varied order, but similar sequence, and multiplied by
2, 5, 6, 7, 8, or 11, it yields a different series, with similar
characteristics. Thus:--
76923 × 1 = 76923
× 3 = 230769
× 4 = 307692
× 9 = 692307
× 10 = 769230
× 12 = 923076
76923 × 2 = 153846
× 5 = 384615
× 6 = 461538
× 7 = 538461
× 8 = 615384
× 11 = 846153
DON’T BUY IT TO TRY IT
A kaleidoscope cylinder contains twenty small pieces of coloured glass.
As we turn it round, or shake it, so as to make ten changes of pattern
every minute, it will take the inconceivable space of time of
462,880,899,576 years and 360 days to exhaust all the possible
symmetrical variations. (The 360 days is good!)
No. LXXXI.--PINS AND DOTS
Here is an amusing little exercise for the ingenuity of our solvers.
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
| \ / | \ / | \ / | \ / | \ / |
| \ / | \ / | \ / | \ / | \ / |
| * | * | * | * | * |
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
*-----------*-----------*-----------*-----------*-----------*
Take six sharp pins, and puzzle out how to stick them into six of the
black dots, so that no two pins, are on the same line, in any direction,
vertical, horizontal, or diagonal.
No. LXXXII.--A TRICKY COURSE
The middle of a large playground was paved with sixty-four square
flagstones of equal size, which are numbered on this diagram from one to
sixty-four.
+----+----+----+----+----+----+----+----+
| _1_| _9_|_17_|_25_|_33_|_41_|_49_|_57_|
+----+----+----+----+----+----+----+----+
| _2_|_10_|_18_|_26_|_34_|_42_|_50_|_58_|
+----+----+----+----+----+----+----+----+
| _3_|_11_|_19_|_27_|_35_|_43_|_51_|_59_|
+----+----+----+----+----+----+----+----+
| _4_|_12_|_20_|_28_|_36_|_44_|_52_|_60_|
+----+----+----+----+----+----+----+----+
| _5_|_13_|_21_|_29_|_37_|_45_|_53_|_61_|
+----+----+----+----+----+----+----+----+
| _6_|_14_|_22_|_30_|_38_|_46_|_54_|_62_|
+----+----+----+----+----+----+----+----+
| _7_|_15_|_23_|_31_|_39_|_47_|_55_|_63_|
+----+----+----+----+----+----+----+----+
| _8_|_16_|_24_|_32_|_40_|_48_|_56_|_64_|
+----+----+----+----+----+----+----+----+
One of the schoolmasters, who had a head for puzzles, took his stand
upon the square here numbered 19, and offered a prize to any boy who,
starting from the square numbered 46, could make his way to him, passing
through every square once, and only once. It was after many vain
attempts that the course was at last discovered. Can you work it out?
No. LXXXIII.--FOR THE CHILDREN
Place twelve draughtsmen, or buttons, in a square, so that you count
four along each side of it, thus:--
● ● ● ●
● ●
● ●
● ● ● ●
Now take the same men or buttons, and arrange them so that they form
another square, and you can count five along each side of it.
A GAME OF NINES
Here is a good specimen of the eccentricities and powers of numbers:--
153846 × 13 = 1999998
230769 × 13 = 2999997
307692 × 13 = 3999996
384615 × 13 = 4999995
461538 × 13 = 5999994
538461 × 13 = 6999993
615384 × 13 = 7999992
692307 × 13 = 8999991
No. LXXXIV.--TELL-TALE TABLES
She was quite an old maid, and her age was a most absolute secret.
Determined to discover it, her scapegrace nephew, on Christmas Eve,
produced these tables, and asked her with well simulated innocence on
which of them she could see the number of her age.
+--+--+--+ +--+--+--+ +--+--+--+
| 1|23|45| | 2|23|46| |16|27|54|
+--+--+--+ +--+--+--+ +--+--+--+
| 3|25|47| | 3|26|47| |17|28|55|
+--+--+--+ +--+--+--+ +--+--+--+
| 5|27|49| | 6|27|50| |18|29|56|
+--+--+--+ +--+--+--+ +--+--+--+
| 7|29|51| | 7|30|51| |19|30|57|
+--+--+--+ +--+--+--+ +--+--+--+
| 9|31|53| |10|31|54| |20|31|58|
+--+--+--+ +--+--+--+ +--+--+--+
|11|33|55| |11|34|55| |21|48|59|
+--+--+--+ +--+--+--+ +--+--+--+
|13|35|57| |14|35|58| |22|49|60|
+--+--+--+ +--+--+--+ +--+--+--+
|15|37|59| |15|38|59| |23|50|61|
+--+--+--+ +--+--+--+ +--+--+--+
|17|39|61| |18|39|62| |24|51|62|
+--+--+--+ +--+--+--+ +--+--+--+
|19|41| | |19|42| | |25|52| |
+--+--+ A| +--+--+ B| +--+--+ C|
|21 43| | |22 43| | |26|53| |
+--+--+--+ +--+--+--+ +--+--+--+
+--+--+--+ +--+--+--+ +--+--+--+
| 8|27|46| | 4|23|46| |32|43|54|
+--+--+--+ +--+--+--+ +--+--+--+
| 9|28|47| | 5|28|47| |33|44|55|
+--+--+--+ +--+--+--+ +--+--+--+
|10|29|56| | 6|29|52| |34|45|56|
+--+--+--+ +--+--+--+ +--+--+--+
|11|30|57| | 7|30|53| |35|46|57|
+--+--+--+ +--+--+--+ +--+--+--+
|12|31|58| |12|31|54| |36|47|58|
+--+--+--+ +--+--+--+ +--+--+--+
|13|40|59| |13|36|55| |37|48|59|
+--+--+--+ +--+--+--+ +--+--+--+
|14|41|60| |14|37|60| |38|49|60|
+--+--+--+ +--+--+--+ +--+--+--+
|15|42|61| |15|38|61| |39|50|61|
+--+--+--+ +--+--+--+ +--+--+--+
|24|43|62| |20|39|62| |40|51|62|
+--+--+--+ +--+--+--+ +--+--+--+
|25|44| | |21|44| | |41|52| |
+--+--+ D| +--+--+ E| +--+--+ F|
|26|45| | |22|45| | |42|53| |
+--+--+--+ +--+--+--+ +--+--+--+
From her answer he was able to calculate that the old lady was
fifty-five.
The tell-tale tables disclosed her age thus:--As it appeared in tables
A, B, C, E, and F, he added together the numbers at the top left-hand
corners, and found the total to be fifty-five. This rule applies in all
cases.
No. LXXXV.--A PAPER PUZZLE
Of the many paper-cutting tricks which appeal to us none is more simple
and attractive than this:--
+---+
| |
| |
+-----+ +-----+
| |
+-----+ +-----+
/ \ | | / \
\ / | | \ /
+-+ | | +-+
| | | | | |
| | | | | |
| | | | | |
| | +---+-+-+---+ | |
+-+ | | | +-+
+---+ | +-+-+ | +---+
| +---+---+-+-+---+---+ |
| | |
+-------------+-------------+
Take a piece of paper, say 5 inches by 3 inches, but any oblong shape
and size will do, and after folding it four times cut it lengthways up
the centre. Unfold the pieces, and to your surprise you will find a
perfect cross and other pieces in pairs of the shapes shown above. The
puzzle is how to fold the paper.
+--------------+----------+
|b d¦ |
| ¦ |
| ¦ |
| e¦ |
| ¦ |
| ¦ |
|a c¦ |
+--------------+----------+
The paper must be folded first so that B comes upon C, then so that A
comes upon D, then from D to C, and lastly from E to C. If it is now cut
lengthways exactly along the centre the figures shown on the original
diagram will be formed, which resemble a cross and lighted candles on an
altar.
No. LXXXVI.--A HOME-MADE PUZZLE
Take a thin board, about eight inches square, and mark it out into
thirty-six equal parts; bore a hole in the centre of each part, and then
fit in a small wooden peg, leaving about a quarter inch above the
surface, as is shown in Fig. 1, the section below the diagram.
[Illustration]
Prepare thirty-six pieces of white or coloured cardboard of the length A
to B, and place them over the pegs in any direction in which they will
fit so as to form some such symmetrical pattern as is given on the
second diagram, putting two holes only on each peg. Chess-players will
see that this is the regular knight’s move.
[Illustration]
Quite a number of beautiful designs can be thus formed, and those who
have not the means at hand for making a complete set can enjoy the
puzzle by merely marking out thirty-six squares, and drawing lines from
centre to centre of the exact length from A to B, with black or coloured
pencils.
No. LXXXVII.--LOYD’S MITRE PROBLEM
[Illustration]
Divide this figure into four similar and equal parts.
A PRETTY PROBLEM
The solution of the pretty little problem: place three twos in three
different groups, so that twice the first group, or half the third group
equals the second group, is this:--
2 1 2 2 + 2
----- = - 2 - - = 1 ----- = 2
2 + 2 2 2 2
No. LXXXVIII.--CONTINUOUS LINES
The following figure, which represents part of a brick wall, cannot be
marked out along all the edges of the bricks in less than six continuous
lines without going more than once over the same line:--
[Illustration]
Here, in strong contrast to the simple figure given above, which could
not be traced without lifting the pen six times from the paper, is an
intricate design, the lines of which, on the upper or on the lower half,
can be traced without any break at all.
[Illustration]
The general rule that governs such cases is, that where an uneven number
of lines meet a fresh start has to be made. In the diagram now given the
only such points are at the extremities of the upper and lower halves of
the figure at A and X. At all other points two, or four, or six lines
converge, and there is no break of continuity in a tracing of the
figure.
No. LXXXIX.--CUT OFF THE CORNERS
Can you suggest quite a simple and practical way to fix the points on
the sides of a square which will be at the angles of an octagon formed
by cutting off equal corners of the square, as shown below?
A E F B
+------+--------+------+
| / \ |
| / \ |
|/ \|
M+ +G
| |
| |
| |
L+ +H
|\ /|
| \ / |
| \ / |
+------+--------+------+
C K I D
MYSTIC FIGURES
Very interesting and curious are the properties of the figures 142857,
used in varied order but always in similar sequence, in connection with
7 and 9:--
142857 × 7 = 999999 ÷ 9 = 111111
285714 × 7 = 1999998 ÷ 9 = 222222
428571 × 7 = 2999997 ÷ 9 = 333333
571428 × 7 = 3999996 ÷ 9 = 444444
714285 × 7 = 4999995 ÷ 9 = 555555
857142 × 7 = 5999994 ÷ 9 = 666666
No. XC.--THE FIVE TRIANGLES
The subjoined diagram shows how a square with sides that measure each 12
yards can be divided into five triangles, no two of which are of equal
area, and of which the sides and areas can be expressed in yards by
whole numbers:--
[Illustration]
The areas of these triangles are 6, 12, 24, 48, and 54 square yards
respectively, and the sum of these, 144 square yards, is the area of the
square.
CURIOUS COINCIDENCES
Our readers may remember the remarkable fact that the figures of the
sum, £12, 12s. 8d., when written thus, 12,128, exactly represent the
number of farthings it contains. Now this, so far as we know, is the
only instance of the peculiarity, but there are at least five other
cases which come curiously near to it. They are these:--
£ s. d.
9 9 6 = 9096 farthings
6 6 4 = 6064 „
3 3 2 = 3032 „
10 10 6¹⁄₂ = 10106 „
13 13 8¹⁄₂ = 13138 „
No. XCI.--PLACING A LADDER
If a ladder, with rungs 1 foot apart, rests against a wall, and its
thirteenth rung is 12 feet above the ground, the foot of the ladder is
25 feet from the wall.
[Illustration]
_Proof._--Drop a perpendicular from A to B. Then, as A B C is a right
angle, and the squares on A C, A B, are 169 feet and 144 feet, the
square on C B must be 25 feet, and the length of C B is 5 feet. We thus
move 5 feet towards the wall in going 13 feet up the ladder, and in
mounting 65 feet (five times as far) we must cover 25 feet.
No. XCII.--GRACEFUL CURVES
A prettily ingenious method of dividing the area of a circle into
quarters, each of them a perfect curve, with perimeter (or enclosing
line) equal to the circumference of the circle, and with which four
circles can be formed, is clearly shown by the subjoined diagrams:--
[Illustration]
NIGHTS AT A ROUND TABLE
The host of a large hotel at Cairo noticed that his Visitors’ Book
contained the names of an Austrian, a Brazilian, a Chinaman, a Dane, an
Englishman, a Frenchman, a German, and a Hungarian. Moved by this
curious alphabetical list, he offered them all free quarters and the
best of everything if they could arrange themselves at a round
dining-table so that not one of them should have the same two neighbours
on any two occasions for 21 successive days.
The following is one of many ways in which this arrangement can be made,
and it seems to be the simplest of them all.
Number the persons 1 to 8; and for our first day set them down in
numerical order _except that the two centre ones (4 and 5) change
places_:
(1st day)--1 2 3 5 4 6 7 8
Keep the 1 and the 7 unaltered but double each of the other numbers.
When the product is greater than 8, divide by 7, and only set down the
remainder. Thus we get:
(8th day)--1 4 6 3 8 5 7 2
(Here the fourth figure 3 is 5 × 2 ÷ 7, giving _remainder_ 3, and so
on.)
Repeat this operation once more:
(15th day)--1 8 5 6 2 3 7 4
To fill in the intermediate days we have only to keep 1 unchanged and
let the remaining numbers run downwards in _simple numerical order_,
following 8 with 2, 2 with 3, and so on. Thus:--
1st day--1 2 3 5 4 6 7 8
2nd day--1 3 4 6 5 7 8 2
3rd day--1 4 5 7 6 8 2 3
4th day--1 5 6 8 7 2 3 4
5th day--1 6 7 2 8 3 4 5
6th day--1 7 8 3 2 4 5 6
7th day--1 8 2 4 3 5 6 7
-------------------------
8th day--1 4 6 3 8 5 7 2
9th day--1 5 7 4 2 6 8 3
10th day--1 6 8 5 3 7 2 4
11th day--1 7 2 6 4 8 3 5
12th day--1 8 3 7 5 2 4 6
13th day--1 2 4 8 6 3 5 7
14th day--1 3 5 2 7 4 6 8
-------------------------
15th day--1 8 5 6 2 3 7 4
16th day--1 2 6 7 3 4 8 5
17th day--1 3 7 8 4 5 2 6
18th day--1 4 8 2 5 6 3 7
19th day--1 5 2 3 6 7 4 8
20th day--1 6 3 4 7 8 5 2
21st day--1 7 4 5 8 2 6 3
This completes the schedule. It will be found on examination that every
number is between every pair of the other numbers once, and once only.
In order to reduce our first-day ring to exact numerical order we have
only to interchange the numbers 4 and 5 throughout. The first three
lines for example would then become:
1 2 3 4 5 6 7 8
1 3 5 6 4 7 8 2
1 5 4 7 6 8 2 3, etc.
or, by putting letters for figures,
A B C D E F G H
A C E F D G H B
A E D G F H B C, etc.
An arrangement of the guests is thus arrived at for twenty-one
successive days, so that not one of them has the same two neighbours on
any two occasions.
No. XCIII.--MAKING MANY SQUARES
Can you apply the two oblongs drawn below to the two concentric squares,
so as to produce thirty-one perfect squares?
+------+ +------+
| | | |
| | | |
+----------------------------------+ +------+ +------+
| | | | | |
| | | | | |
| +--------------------+ | +------+ +------+
| | | | | | | |
| | | | | | | |
| | | | +------+ +------+
| | | | | | | |
| | | | | | | |
| | | | +------+ +------+
| | | | | | | |
| | | | | | | |
| +--------------------+ | +------+ +------+
| | | | | |
| | | | | |
+----------------------------------+ +------+ +------+
| | | |
| | | |
+------+ +------+
No. XCIV.--CUT ACROSS
Take a piece of cardboard in the form of a Greek cross with arms, as
shown here, and divide it by two straight cuts, so that the pieces when
reunited form a perfect square.
+------+
| |
| |
+------+ +------+
| |
| |
+------+ +------+
| |
| |
+------+
No. XCV.--A PRETTY PUZZLE
The diagrams which we give below show how a hollow square can be formed
of the pieces of three-quarters of another square from which a corner
has been cut away:--
+-------+ +-------+ +-------+-------+
| | | | | | |
| | | +---+ | +---+---+ |
| | | | | | | | |
| +-------+ +---+ +---+---+ +---+ +---+
| | | | | | | | | |
| | | +---+---+ | | +---+---+ |
| | | | | |
+---------------+ +---------------+ +---------------+
No. XCVI.--A FIVE-FOLD SQUARE
The subjoined diagram shows how a square of paper or cardboard may be
cut into nine pieces which, when suitably arranged, form five perfect
squares.
[Illustration]
THE SOCIABLE SCHOOLGIRLS
On how many days can fifteen schoolgirls go out for a walk so arranged
in rows of three, that no two are together more than once?
Fifteen schoolgirls can go out for a walk on seven days so arranged in
rows of three that no two are together more than once.
It is said, on high authority, that there are no less than
15,567,522,000 different solutions to this problem. Here is one of them,
given in _Ball’s Mathematical Recreations_, in which _k_ stands for one
of the girls, and _a_, _b_, _c_, _d_, _e_, _f_, _g_, in their
modifications, for her companions on the seven different days:--
+---------+---------+---------+---------+---------+---------+--------+
| Sunday | Monday | Tuesday |Wednesday| Thursday| Friday |Saturday|
+---------+---------+---------+---------+---------+---------+--------+
|_ka₁a₂_ |_kb₁b₂_ |_kc₁c₂_ |_kd₁d₂_ |_ke₁e₂_ |_kf₁f₂_ |_kg₁g₂_ |
|_b₁d₁f₁_ |_a₁d₂e₂_ |_a₁d₁e₁_ |_a₂b₂c₂_ |_a₂b₁c₁_ |_a₁b₂c₁_ |_a₁b₁c₂_|
|_b₂e₁g₁_ |_a₂f₂g₂_ |_a₂f₁g₁_ |_a₁f₂g₁_ |_a₁f₁g₂_ |_a₂d₂e₁_ |_a₂d₁e₂_|
|_c₁d₂g₂_ |_c₁d₁g₁_ |_b₁d₂f₂_ |_b₁e₁g₂_ |_b₂d₁f₂_ |_b₁e₂g₁_ |_b₂d₂f₁_|
|_c₂e₂f₂_ |_c₂e₁f₁_ |_b₂e₂g₂_ |_c₁e₂f₁_ |_c₂d₂g₁_ |_c₂d₁g₂_ |_c₁e₁f₂_|
+---------+---------+---------+---------+---------+---------+--------+
It is an excellent game of patience, for those who have time and
inclination, to place the figures 1 to 15 inclusive in seven such
columns, so as to fulfil the conditions.
No. XCVII.--THE THREE CROSSES
It is possible from a Greek cross to cut off four equal pieces which,
when put together, will form another Greek cross exactly half the size
of the original, and by this process to leave a third Greek cross
complete.
[Illustration]
This is how to do it:--
Bisect _C D_ at _N_, _F G_ at _O_, _K L_ at _P_, and _B I_ at _Q_.
Join _N H_, _O M_, _P A_, _Q E_, intersecting at _R_, _S_, _T_, _U_.
Bisect _A R_ at _V_, _E S_ at _W_, _T H_ at _X_, and _M U_ at _Y_.
Join _V Q_, _N Y_, _W P_, _O X_, _N W_, _V O_, _Q X_, _Y P_.
Carefully cut out from the original Greek cross the newly-formed Greek
cross, and the odd pieces from around it can be arranged to form another
Greek cross.
No. XCVIII.--THE HAMMOCK
The greatest number of plane figures that can be formed by the union of
ten straight lines is thirty-six.
[Illustration]
The two equal lines at right angles are first drawn, and each is divided
into eight equal parts. The other eight straight lines are then drawn
from _a_ to _a_, from _b_ to _b_, and so on, until the hammock-shaped
network of thirty-six plane figures is produced.
No. XCIX.--CUTTING A CRUMPET
It will be seen on the diagram below that seven straight vertical cuts
with a table-knife will divide a crumpet into twenty-eight parts.
[Illustration]
BALANCE THE SCALES
The nine digits can be so adjusted as to form an equation, or, if taken
as weight, to balance the scales. Thus:--
9, 6¹⁄₂ = 3, 5, 7⁴⁄₈
TRUE STRETCHES OF IMAGINATION
How large is the sea? This is a bold big question, and any possible
answer involves a considerable stretch of the imagination. Here is a
startling illustration of its vast volume:--
If the water of the sea could be gathered into a round column, reaching
the 93,000,000 miles which separate us from the sun, the diameter of
this column would be nearly two miles and a half!
It is perhaps even more difficult to realise that this mighty mass of
waters could be dissipated in a few moments, if the column we have
imagined could become ice, and if the entire heat of the sun could be
concentrated upon it. All would be melted in _one second of time_, and
converted into steam in eight seconds!
No. C.--TO MAKE AN ENVELOPE
Many of our readers may be glad to know an easy way to make an envelope
of any shape or size.
[Illustration]
This diagram speaks for itself. When the lines _A B_, _A C_, _D B_, _D
C_ have been drawn, the corners of the rectangle, _H E G F_, are folded
over, as shown by the dotted lines, after the corners have been rounded,
and the margins touched with gum.
No. CI.--SQUARING AN OBLONG
The diagrams below will show how a piece of paper, 15 inches long and 3
inches wide, can be cut into five parts, and rearranged to form a
perfect square.
[Illustration]
A PLAGUE OF BLOW-FLIES
The following astounding calculation is answer enough to a question put
by one of the authors of “Rejected Addresses:”--“Who filled the
butchers’ shops with big blue flies?”
A pair of blow-flies can produce ten thousand eggs, which mature in a
fortnight. If every egg hatches out, and there are equal numbers of
either sex, which forthwith increase and multiply at the same rapid
rate, and if their descendants do the like, so that all survive at the
end of six months, it has been calculated that, if thirty-two would fill
a cubic inch of space, the whole innumerable swarm would cover the
globe, land and sea, half a mile deep everywhere.
No. CII.--BY RULE OF THUMB
Here is quite a neat way to make an equilateral triangle without using
compasses:--
[Illustration]
Take a piece of paper exactly square, which we will call _A B C D_, fold
it across the middle, so as to form the crease _E F_; unfold it, and
fold it again so that the corner _D_ falls upon the crease _E F_ at _G_,
and the angle at _G_ is exactly divided. Again unfold the square, and
from _G_ draw the straight lines _G C_ and _G D_. Then _G C D_ is the
equilateral triangle required.
THE VALUE OF A FRENCHWOMAN
How can we be sure that the value of a Frenchwoman is just 1 franc 8
centimes?
We can be sure that the exact value of a Frenchwoman is 1 franc 8
centimes, for
Two Frenchwomen = Deux Françaises.
Deux Françaises = deux francs seize.
(2 francs 16).
Therefore, One Frenchwoman = 1 franc 8!
No. CIII.--CLEARING THE WALL
If a 52-feet ladder is set up so as just to clear a garden wall 12 feet
high and 15 feet from the building, it will touch the house 48 feet from
the ground.
[Illustration]
Our diagram shows this, and also, by a dotted line, the only other
possible position in which it could fulfil the conditions, if it were
then of any practical use.
A BURDEN OF PINS
If one pin could be dropped into a vessel this week, two the next, four
the next, and so on, doubling each time for a year, the accumulated
quantity would be 4,503,599,627,370,495, and their weight, if we reckon
200 pins to the ounce, would amount to 628,292,358 tons, a full load for
27,924 ships as large as the _Great Eastern_, whose capacity was 22,500
tons.
No. CIV.--MEMORIES OF EUCLID
When at the signpost which said “To _A_ 4 miles, to _B_ 9 miles” on one
arm, and on the other “To _C_ 3 miles, to _D_ ---- miles,” and the boy
whom I met could only tell me that the farm he worked at was equidistant
from _A_, _B_, _C_, and _D_, and nearer to them than to the signpost,
and that all the roads ran straight, I found, thanks to memories of
Euclid, that I was 12 miles from _D_.
[Illustration]
Since _B A_ and _D C_ intersect outside the circle at the signpost _E_,
therefore _A E_ × _E B_ = _C E_ × _E D_.
but _A E_ × _E B_ = 4 × 9 = 36,
therefore _C E_ × _E D_ = 36,
and _C E_ = 3, therefore _E D_ = 12.
_Q.E.D._
No. CV.--A TRANSFORMATION
This seems to be quite a poor attempt at a Maltese cross, but there is
method in the madness of its make.
[Illustration]
It is possible by two straight cuts to divide this uneven cross into
four pieces which can be arranged together again so that they form a
perfect square. Where must the cuts be made, and how are the four pieces
rearranged?
BREVITY IS THE SOUL OF WIT
We all remember that splendidly terse message of success sent home to
the authorities by Napier when he had conquered the armies of
Scinde--“Peccavi!” (I have sinned).
History had an excellent opportunity for repeating itself when Admiral
Dewey defeated the Spaniel fleet, for he might have conveyed the news of
his victory by the one burning word--“Cantharides”--“The Spanish fly!”
No. CVI--SHIFTING THE CELLS
In the diagram below a square is subdivided into twenty-five cells.
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
|/ \|/ \|/ \|/ \|/ \|
| | | | | |
| | | | | |
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
|/ \|/ \|/ \|/ \|/ \|
| | | | | |
| | | | | |
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
|/ \|/ \|/ \|/ \|/ \|
| | | | | |
| | | | | |
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
|/ \|/ \|/ \|/ \|/ \|
| | | | | |
| | | | | |
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ | / \ |
| / \ | / \ | / \ | / \ | / \ |
|/ \|/ \|/ \|/ \|/ \|
| | | | | |
| | | | | |
+-----------+-----------+-----------+-----------+-----------+
Can you, keeping always on the straight lines, cut this into four
pieces, and arrange these as two perfect squares, in which every
semicircle still occupies the upper half of its cell?
A HOME-MADE MICROSCOPE
The following very simple recipe for a home-made microscope has been
suggested by a Fellow of the Royal Microscopical Society:--
Take a piece of black card, make a small pinhole in it, put it close to
the eye, and look at some small object closely, such as the type of a
newspaper. A very decided magnifying power will be shown thus.
No. CVII.--IN A TANGLE
Where on this diagram must we place twenty-one pins or dots so that they
fall into symmetrical design and form thirty rows, with three in each
row?
[Illustration]
ON ALL FOURS
Those who are fond of figures will find it a most interesting exercise
to see how far they are able to represent every number, from one up to a
hundred, by the use of four fours. Any of the usual signs and symbols of
arithmetic may be brought into use. Here are a few instances of what may
thus be done:--
4 + 4 + 4 4
3 = ---------; 9 = 4 + 4 + -;
4 4
4
36 = 4(4 + 4) + 4; 45 = 44 + -;
4
52 = 44 + 4 + 4; 60 = 4 × 4 × 4 - 4.
No. CVIII.--STILL A SQUARE
This figure, which now forms a square, and the quarter of that square,
can be so divided by two straight lines that its parts, separated and
then reunited, form a perfect square. How is this done?
+------+------+------+
| | | |
| | | |
+------+------+------+
| | |
| | |
+------+------+
A MAGIC SQUARE
Here we have arranged five rows of five cards each, so that no two
similar cards are in the same lines. Counting the ace as eleven, each
row, column, and diagonal adds up to exactly twenty-six.
/-------\ /-------\ /-------\ /-------\ /-------\
| ♣ | | ♥ | | ♠ ♠ | | ♥ ♥ | | ♣ ♣ |
| | | | | | | ♥ | | ♣ ♣ |
| | | ♥ | | ♠ | | ♥ | | ♣ |
| | | | | | | ♥ | | ♣ ♣ |
| ♣ | | ♥ | | ♠ ♠ | | ♥ ♥ | | ♣ ♣ |
\-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\
| ♦ ♦ | | ♠ ♠ | | ♦ ♦ | | | | ♦ |
| | | ♠ | | ♦ ♦ | | | | |
| ♦ | | ♠ | | ♦ ♦ | | ♠ | | ♦ |
| | | ♠ | | ♦ ♦ | | | | |
| ♦ ♦ | | ♠ ♠ | | ♦ ♦ | | | | ♦ |
\-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\
| ♥ ♥ | | | | ♠ | | ♥ ♥ | | ♣ ♣ |
| ♥ ♥ | | | | | | | | ♣ |
| ♥ | | ♣ | | ♠ | | ♥ | | ♣ ♣ |
| ♥ ♥ | | | | | | | | ♣ |
| ♥ ♥ | | | | ♠ | | ♥ ♥ | | ♣ ♣ |
\-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\
| ♣ | | ♥ ♥ | | ♣ ♣ | | ♦ ♦ | | |
| | | | | ♣ | | ♦ ♦ | | |
| ♣ | | ♥ ♥ | | ♣ | | ♦ | | ♥ |
| | | | | ♣ | | ♦ ♦ | | |
| ♣ | | ♥ ♥ | | ♣ ♣ | | ♦ ♦ | | |
\-------/ \-------/ \-------/ \-------/ \-------/
/-------\ /-------\ /-------\ /-------\ /-------\
| ♦ ♦ | | ♠ ♠ | | | | | | ♣ ♣ |
| ♦ | | ♠ ♠ | | | | ♠ ♠ | | |
| ♦ | | ♠ | | ♦ | | | | ♣ |
| ♦ | | ♠ ♠ | | | | ♠ ♠ | | |
| ♦ ♦ | | ♠ ♠ | | | | | | ♣ ♣ |
\-------/ \-------/ \-------/ \-------/ \-------/
After you have looked at this Magic Square, and set it out on the table,
shuffle the cards, and try to re-arrange them so as to give the same
results.
No. CIX.--A TRANSFORMATION
Cut a square of paper or cardboard into seven such pieces as are marked
in this diagram.
[Illustration]
Can you rearrange them so that they form the figure 8.
A SECRET REVEALED
There are several ways in which strange juggling with figures and
numbers is to be done, but none is more curious than this:--
Ask someone, whose age you do not know, to write down secretly the date
and month of his birth in figures, to multiply this by 2, to add 5, to
multiply by 50, to add his age last birthday and 365. He then hands you
_this total only_ from this you subtract 615. This reveals to you at a
glance his age and birthday.
Thus, if he was born April 7 and is 23, 74 (the day and the month) × 2 =
148; 148 + 5 = 153; 153 × 50 = 7650; 7650 + 23 (his age) = 7673; and
7673 + 365 = 8038. If from this you subtract 615, you have 7423, which
represents to you the _seventh day of the fourth month, 23 years age_!
This rule works out correctly in all cases.
No. CX.--TO MAKE AN OBLONG
Cut out in stiff paper or cardboard two pieces of the shape and size of
the small triangle, and four pieces of the shapes and sizes of the other
three patterns--fourteen pieces in all.
[Illustration]
The puzzle is to fit these pieces together so that they form a perfect
oblong.
SLEEPERS THAT SLIP AND SLEEP
Here is a string of sentences, which may be used as stimulating mental
gymnastics when we leave the “Land of Nod.”
A sleeper runs on sleepers, and in this sleeper on sleepers sleepers
sleep. As this sleeper carries its sleepers over the sleepers that are
under the sleeper, a slack sleeper slips. This jars the sleeper and its
sleepers, so that they slip and no longer sleep.
DAYS THAT ARE BARRED
Clever calculation has established a fact which we shall not be able to
verify by personal experience. Whatever else may happen, the first day
of a century can never fall on Sunday, Wednesday, or Friday.
No. CXI.--SQUARES ON THE CROSS
On this cross there are seventeen distinct and perfect squares marked
out at their corners by asterisks.
*-----------*
| |
| |
| |
| |
| |
*-----------*
/| \ / |\
/ | \ / | \
/ | * | \
/ | / \ | \
/ | / \ | \
*-----------*-----------*-----------*-----------*-----------*
| | \ / | \ / | \ / | |
| | \ / | \ / | \ / | |
| | * | * | * | |
| | / \ | / \ | / \ | |
| | / \ | / \ | / \ | |
*-----------*-----------*-----------*-----------*-----------*
\ | \ / | /
\ | \ / | /
\ | * | /
\ | / \ | /
\| / \ | /
*-----------*
| |
| |
| |
| |
| |
*-----------*
How few, and which, of these can you remove, so that not a single
perfect square remains?
STANDING ROOM FOR ALL
On a globe 2 feet in diameter the Dead Sea appears but as a small
coloured dot. If it were frozen over there would be standing room on its
surface for the whole human race, allowing 6 square feet for each
person; and if they were all suddenly engulfed, it would merely raise
the level of the lake about 4 inches.
No. CXII.--A CHINESE PUZZLE
Here is quite a good exercise for ingenious brains and fingers. Cut a
piece of stiff paper or cardboard into such a right-angled triangle as
is shown below.
+
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
+----------------------------+
+----+
| |
+--+ +--+
| |
+--+ +--+
| |
+--+ +--+
| |
+--+ +--+
| |
+--+ +--+
| |
+--+ +--+
| |
+----+
Can you divide this into only three pieces, which, when rearranged, will
form the design given as No. 2?
A DEAL IN ROSES
I sent an order for dwarf roses to a famous nursery-garden, asking for a
parcel of less than 100 plants, and stipulating that if I planted them 3
in a row there should be 1 over; if 4 in a row 2 over; if 5 in a row 3
over, and if 6 in a row 4 over, as a condition for payment.
The nurseryman was equal to the occasion, charged me for 58 trees, and
duly received his cheque.
No. CXIII.--FIRESIDE FUN
This puzzle is not so easy of solution as it may seem at first sight.
Take a counter, or a coin, and place it on one of the points; then push
it across to the opposite point, and leave it there. Do this with a
second counter or coin, starting on a vacant point, and continue this
process until every point is covered, as we place the eighth counter or
coin on the last point.
A+ +B
| \ / |
| \ / |
| + |
| / \ |
| / \ |
H+-----------+-----------+-----------+C
\ / | | \ /
\ / | | \ /
+ | | +
/ \ | | / \
/ \ | | / \
G+-----------+-----------+-----------+D
| \ / |
| \ / |
| + |
| / \ |
| / \ |
F+ +E
A NOVEL EXERCISE
“Write down,” said a schoolmaster, “the nine digits in such order that
the first three shall be one third of the last three, and the central
three the result of subtracting the first three from the last.”
The arrangement which satisfies these conditions is, 219, 438, 657.
CURIOUS CALCULATIONS
1. There is a sum of money of such sort that its pounds, shillings, and
pence, written down as one continuous number, represent exactly the
number of farthings which it contains. What is it?
THE SLIP CARRIAGE
2. If on a level track a train, running all the time at 30 miles an
hour, slips a carriage, which is uniformly retarded by the brakes, and
which comes to rest in 200 yards, how far has the train itself then
travelled?
3. A traveller said to the landlord of an inn, “Give me as much money as
I have in my hand, and I will spend sixpence with you.” This was done,
and the process was twice repeated, when the traveller had no money
left. How much had he at first?
4. How can we obtain eleven by adding one-third of twelve to four-fifths
of seven?
FILL IN THE GAPS
5. Can you replace the missing figures in this mutilated long division
sum?
215)*7*9*(1**
***
-----
*5*9
*5*5
-----
*4*
***
=====
6. I buy as many heads of asparagus in the market as can be contained by
a string 1 foot long. Next day I take a string 2 feet long, and buy as
many as it will gird, offering double the price that I have given
before. Was this a reasonable offer?
ALIKE FROM EITHER END
7. As “one good turn deserves another,” first reverse me, then reverse
my square, then my cube, then my fourth power. When all this is done no
change has been made. What am I?
THE SEALED BAGS
8. How can a thousand pounds be so conveniently stored in ten sealed
bags that any sum in pounds from £1 to £1000 can be paid without
breaking any of the seals?
9. This is at once a problem and a puzzle:--
Though you twist and turn me over,
Yet no change can you discover.
Take me thrice, and cut in twain,
You will find but one remain.
10. Three gamblers, when they sit down to play, agree that the loser
shall always double the sum of money that the other two have before
them. After each of them has lost once, it is found that each has eight
sovereigns on the table. How much money had each at starting?
FIND THE MULTIPLIER
11. When Tom’s back was turned, the boy sitting next to him rubbed out
almost all his sum. Tom could not remember the multiplier, and only
this remained on his slate--
345
..
-----
....
....
-----
..76.
Can you reconstruct the sum?
JUGGLING WITH THE DIGITS
12. The sum of the nine digits is 45. Can you hit upon other
arrangements of 1, 2, 3, 4, 5, 6, 7, 8, 9, writing each of them once
only, which will produce the same total. Of course fractions may be
used.
A BRAIN TWISTER
13. The combined ages of Mary and Ann are forty-four years. Mary is
twice as old as Ann was when Mary was half as old as Ann will be when
Ann is three times as old as Mary was when Mary was three times as old
as Ann. How old, then, is Mary?
14. Mr Oldboy was playing backgammon with his wife on the eve of his
golden wedding, and could not make up his mind whether he should leave a
blot where it could be taken up by an ace, or one which a tré would hit.
His grandson, at home from Cambridge for the Christmas vacation, solved
the question for him easily. What was his decision?
“ASK WHERE’S THE NORTH?”--_Pope._
15. I am aboard a steamer, anchored in a bay where the needle points due
north, and exactly 1200 miles from the North Pole. If the course is
perfectly clear, and I steam continuously at the rate of 20 miles an
hour, always steering north by the compass needle, how long will it take
me to reach the North Pole?
16. Three persons, _A_, _B_, and _C_, share twenty-one wine casks of
equal capacity, of which seven are full, seven are half full, and seven
are empty. How can these be so apportioned that each person shall have
an equal number of casks, and an equal quantity of wine, without
transferring any of it from cask to cask?
17. A hungry mouse, in search of provender, came upon a box containing
ears of corn. He could carry three ears home at a time, and only had
opportunity to make fourteen journeys to and fro. How many ears could he
add to his store?
18. Take exactly equal quantities of lard and butter; mix a small piece
of the butter intimately with all the lard. From this blend take a piece
just as large as the fragment removed from the butter, and mix this
thoroughly with the butter. Is there now more lard in the butter or more
butter in the lard?
WHERE IS THE FALLACY?
19. Here is an apparent proof that 2 = 3:--
4 - 10 = 9 - 15
4 - 10 + ²⁵⁄₄ = 9 - 15 + ²⁵⁄₄
and the square roots of these:--
2 - ⁵⁄₂ = 3 - ⁵⁄₂
therefore
2 = 3.
PARCEL POST LIMITATIONS
20. Our Parcel Post regulations limit the length of a parcel to 3 feet 6
inches, and the length and girth combined to 6 feet. What is the largest
parcel of any size that can be sent through the post under these
conditions?
21. How can we show, or seem to show, that either four, five, or six
nines amount to one hundred?
TEST YOUR SKILL
22. Can you arrange nine numbers in the nine cells of a square, the
largest number 100, and the least 1, so that the product by
multiplication of each row, column and diagonal is 1000?
CATCHING CRABS
23. Seven London boys were at the seaside for a week’s holiday, and
during the six week-days they caught four fine crabs in pools under the
rocks, when the tide was out at Beachy Head.
Hearing of this, the twenty-one boys of a school in the neighbourhood
determined to explore the pools; but with the same rate of success they
only caught one large crab. For how long were they busy searching under
the seaweed?
SETTING A WATCH
24. On how many nights could a watch be set of a different trio from a
company of fifteen soldiers, and how often on these terms could one of
them, John Pipeclay, be included?
“IMPERIAL CÆSAR”
25. If Augustus Cæsar was born on September 23rd, B.C. 63, on what day
and in what year did he celebrate his sixty-third birthday, and by what
five-letter symbol can we express the date?
26. _A_ was born in 1847, _B_ in 1874. In what years have the same two
digits served to express the ages of both, if they are still living?
THIS SHOULD “AMUSE”
27. A hundred and one by fifty divide,
To this let a cypher be duly applied;
And when the result you can rightly divine,
You find that its value is just one in nine.
OVER THE FERRY
28. A man started one Monday morning with money in his purse to buy
goods in a neighbouring town. He paid a penny to cross the ferry, spent
half of the money he then had, and paid another penny at the ferry on
his return home.
He did exactly the same for the next five days, and on Saturday evening
reached home with just one penny in his pocket. How much had he in his
pocket on Monday before he reached the ferry?
THE MEN, THE MONKEY, AND THE MANGOES
29. Three men gathered mangoes, and agreed that next day they would give
one to their monkey and divide the rest equally. The first who arrived
gave one to the monkey, and then took his proper share; the second came
later and did likewise, and the third later still, neither knowing that
any one had preceded him. Finally they met, and, as there were still
mangoes, gave one to the monkey, and shared the rest equally. How many
mangoes at least must there have been if all the divisions were
accurate?
30. I look at my watch between four and five, and again between seven
and eight. The hands have, I find, exactly changed places, so that the
hour-hand is where the minute-hand was, and the minute-hand takes the
place of the hour-hand. At what time did I first look at my watch?
A LARGE ORDER
31. There is a number consisting of twenty-two figures, of which the
last is 7. If this is moved to the first place, the number is increased
exactly sevenfold. Can you discover this lengthy number?
32. A farmer borrowed from a miller a sack of wheat, 4 feet long and 6
feet in circumference. He sent in repayment two sacks, each 4 feet long
and 3 feet in circumference. Was the miller satisfied?
THE FIVE GAMBLERS
33. Five gamblers, whom we will call _A_, _B_, _C_, _D_, and _E_, play
together, on the condition that after each hazard he who loses shall
give to all the others as much as they then have in hand.
Each loses in turn, beginning with _A_, and when they leave the table
each has the same sum in hand, thirty-two pounds. How much had each at
first?
34. Knowing that the square of 87 is 7569, how can we rapidly, without
multiplication, determine in succession the squares of 88, 89, and 90?
NOT WHOLE NUMBERS
35. Two numbers seek which make eleven,
Divide the larger by the less,
The quotient is exactly seven,
As all who solve it will confess.
AN AMPLE CHOICE
36. If there are twenty sorts of things from which
^99999999999999999999^ different selections can be made, how many of
each sort are there?
37. Three women, with no money, went to market. The first had
thirty-three apples, the second twenty-nine, the third twenty-seven.
Each woman sold the same number of apples for a penny. They all sold
out, and yet each received an equal amount of money. How was this?
SIMPLE SUBTRACTION
38. Take five from five, oh, that is mean!
Take five from seven, and this is seen.
39. If a bun and a half cost three halfpence, how many do you buy for
sixpence?
40. How many times in a day would the hands of a watch meet each other,
if the minute-hand moved backward and the hour-hand forward?
41. How can half-a-crown be equally divided between two fathers and two
sons so that a penny is the coin of smallest value given to them?
SIZE AND SPEED
42. If the number of the revolutions of the wheel of a bicycle in six
seconds is equal to the number of miles an hour at which it is running,
what is the circumference of the wheel?
43. Hearing a clock strike, and being uncertain of the hour, I asked a
policeman. He had a turn for figures, and replied: “Take half, a third,
and a fourth of the hour that has just struck, and the total will be one
larger than that hour.” What o’clock was it?
AFTER LONG YEARS
44. If cash is lent at five per cent.
To those who choose to borrow,
How soon shall I be worth a pound
If I lend a crown to-morrow?
ON THE JUMP
45. If I jump off a table with a 20-lb. dumb-bell in my hand, what is
the pressure upon me of its weight while I am in the air?
AT A BAZAAR
46. In charitable mood I went recently to a bazaar where there were four
tents arranged to tempt a purchaser. At the door of each tent I paid a
shilling, and in each tent I spent half of the money remaining in my
purse, giving the door-keepers each another shilling as I came out.
It took my last shilling to pay the fourth door-keeper. How much money
had I at first, and what did I spend in each tent?
OUT IN THE RAIN
47. Rain is falling vertically, at a speed of 5 miles an hour. I am
walking through it at 4 miles an hour. At what angle to the vertical
must I hold my umbrella, so that the raindrops strike its top at right
angles?
A MONKEY PUZZLER
48. The following interesting problem, translated from the original
Sanscrit, is given by Longfellow in his “Kavanagh”:--
“A tree, 100 cubits high, is distant from a well 200 cubits. From the
top of this tree one monkey descends, and goes to the well. Another
monkey leaps straight upwards from the top, and then descends to the
well by the hypotenuse. Both pass over an equal space. How high does the
second monkey leap?”
49. A steamboat 105 miles east of Tynemouth Lighthouse springs a leak.
She puts back at once, and in the first hour goes at the rate of 10
miles an hour.
More and more water-logged, she decreases her speed each succeeding hour
at the rate of one-tenth of what it has been during the previous hour.
When will she reach the lighthouse?
50. If a hen and a half lays an egg and a half in a day and a half, how
many eggs will twenty-one hens lay in a week?
51. If the population of Bristol exceeds by two hundred and thirty-seven
the number of hairs on the head of any one of its inhabitants, how many
of them at least, if none are bald, must have the same number of hairs
on their heads?
52. A benevolent uncle has in his pocket a sovereign, a half-sovereign,
a crown, a half-crown, a florin, a shilling, and a threepenny piece. In
how many different ways can he tip his nephew, using only these coins,
and how is this most easily determined?
53. Here is a prime problem, in more senses than one, which will tax the
ingenuity of our solvers:--I am a prime number of three figures.
Increased by one-third, ignoring fractions, I become a square number.
Transpose my first two figures and increase me by one-third, and again
I am a square number. Put my first figure last, and increase me by
one-third, and I am another square number. Reverse my three figures, and
increase as before by one-third, and for a fourth time I become a square
number. What are my original figures?
54. In how many different ways can six different things be divided
between two boys?
55. What is quite the highest number that can be scored at six card
cribbage by the dealer, if he has the power to select all the cards, and
to determine the order in which every card shall be played?
COVERING THE WALLS
56. A fanciful collector, who bought pictures with more regard to
quantity than quality, gave instructions that the area of each frame
should exactly equal that of the picture it contained, and that the
frames should be of the same width all round.
At an auction he picked up a so-called “old master,” unframed, which
measured 18 inches by 12 inches. What width of frame will fulfil his
conditions?
A FAMILY REGISTER
57. Our family consists of my mother, a brother, a sister, and myself.
Our average age is thirty-nine. My mother was twenty when I was born; my
sister is two years my junior, and my brother is four years younger than
she is. What are our respective ages?
58. A spider in a dockyard unwittingly attached her web to a mechanical
capstan 1 foot in diameter, at the moment when it began to revolve. To
hold her ground she paid out 73 feet of thread, when the capstan
stopped, and she found herself drained of silk.
To make the best of a bad job she determined to unwind her thread,
walking round and round the capstan at the end of the stretched thread.
When she had gone a mile in her spiral path she stopped, tired and in
despair. How far was she then from the end of her task?
59. A mountebank at a fair had six dice, each marked only on one face 1,
2, 3, 4, 5, or 6, respectively. He offered to return a hundredfold any
stake to a player who should turn up all the six marked faces once in
twenty throws. How far was this from being a fair offer?
CATS’-MEAT FOR DOGS
60. If ninety groats for twenty cats
Will furnish three weeks’ fare,
How many hounds for forty pounds,
Less one, may winter there?
Just ninety days and one assume
The winter’s space to be;
And note that what five cats consume
Will serve for dogs but three.
(A groat = 4d.)
61. Two wineglasses of equal size are respectively half and one-third
full of wine. They are filled up with water, and their contents are then
mixed. Half of this mixture is finally poured back into one of the
wineglasses. What part of this will be wine and what part will be
water?
62. A legend goes that on a stout ship on which St Peter was carried
with twenty-nine others, of whom fourteen were Christians and fifteen
Jews, he so arranged their places, that when a storm arose, and it was
decided to throw half of the passengers overboard, all the Christians
were saved. The order was that every ninth man should be cast into the
sea. How did he place the Christians and the Jews?
A PROLIFIC COW
63. Farmer Southdown was the proud possessor of a prize cow, which had a
fine calf every year for sixteen years. Each of these calves when two
years old, and their calves also in their turn, followed this excellent
example. How many head did they thus muster in sixteen years?
64. A shepherd was asked how many sheep he had in his flock. He replied
that he could not say, but he knew that if he counted them by twos, by
threes, by fours, by fives, or by sixes, there was always one over, but
if he counted them by sevens, there was no remainder. What is the
smallest number that will answer these conditions?
READY RECKONING
65. If a number of round bullets of equal size are arranged in rows one
above another evenly graduated till a single bullet crowns the flat
pyramid, how can their number be readily reckoned, however long the base
line may be?
THE TITHE OF WAR
66. Old General Host
A battle lost,
And reckoned on a hissing,
When he saw plain
What men were slain,
And prisoners, and missing.
To his dismay
He learned next day
What havoc war had wrought;
He had, at most,
But half his host
Plus ten times three, six, ought.
One-eighth were lain
On beds of pain,
With hundreds six beside;
One-fifth were dead,
Captives, or fled,
Lost in grim warfare’s tide.
Now, if you can,
Tell me, my man,
What troops the general numbered,
When on that night
Before the fight
The deadly cannon slumbered?
67. A farmer sends five pieces of chain, of three links each, to be made
into one continuous length. He agrees to pay a penny for each link cut,
and a penny for each link joined. What was the blacksmith entitled to
charge if he worked in the best interest of the _farmer_?
68. In a parcel of old silver and copper coins each silver piece is
worth as many pence as there are copper coins, and each copper coin is
worth as many pence as there are silver coins, and the whole is worth
eighteen shillings. How many are there of each?
A FEAT WITH FIGURES
69. Take the natural numbers 1 to 11, inclusive, and arrange them in
five groups, not using any of them more than once, so that these groups
are equal. Any necessary signs or indices may be used.
HOW OLD WAS JOHN?
70. John Bull passed one-sixth of his life in childhood and one-twelfth
as a youth. When one-seventh of his life had elapsed he had a son who
died at half his father’s age, and John himself lived on four years
more. How old was he at the last?
FIGURE IT OUT
71. There are two numbers under two thousand, such that if unity is
added to each of them, or to the half of each, the result is in every
case a square number. Can you find them?
72. A cheese in one scale of a balance with arms of unequal length seems
to weigh 16 lbs. In the other scale it weighs but 9 lbs. What is its
true weight?
“DIVISION IS AS BAD!”
73. Can you divide 100 into two such parts that if the larger is divided
by the lesser the quotient is also 100?
74. I have marbles in my two side pockets. If I add one to those in the
right-hand pocket, and multiply its increased contents by the number it
held at first, and then deal in a similar way with those in the other
pocket, the difference between the two results is 90. If, however, I
multiply the sum of the two original quantities by the square of their
difference, the result is 176. How many marbles had I at first in each
pocket?
A SURFEIT OF BRIDGE
75. A friendly circle of twenty-one persons agreed to meet each week,
five at a time, for an afternoon of bridge, so long as they could do so
without forming exactly the same party on any two occasions.
As a central room had to be hired, it was important to have some idea as
to the length of time for which they would require it. How long could
they keep up their weekly meetings?
76. A herring and a half costs a penny and a half; what is the price of
a dozen?
77. What sum of money is in any sense seen to be the double of itself?
COMIC ARITHMETIC
78. At the close of his lecture upon unknown quantities, Dr Bulbous
Roots, in playful mood, wrote this puzzle on his blackboard:--
Divide my fifth by my first and you have my fourth; subtract my first
from my fifth and you have my second; multiply my first by my fourth
followed by my second, and you have my third; place my second after my
first and you have my third multiplied by my fourth. What am I?
DROPPED THROUGH THE GLOBE
79. If we can imagine the earth at a standstill for the purpose of our
experiment, and if a perfectly straight tunnel could be bored through
its centre from side to side, what would be the course of a cannon ball
dropped into it from one end, under the action of gravity?
LOVE LETTERS
80. A lady to her lover cried,
“How many notes have you of mine?”
“Six more I’ve sent,” the youth replied,
“Than I have had of thine.”
“But if from one pound ten you take
The pennies we on stamps have spent,
One eighth their cost you thus will make.”
How many had they had and sent?
81. A man, on the day of his marriage, made his will, leaving his money
thus:--If a son should be born, two-thirds of the estate to that son and
one-third to the widow. If a daughter should be born, two-thirds to the
widow and one-third to that daughter. In the course of time twins were
born, a boy and a girl. The man fell sick and died without making a
fresh will. How ought his estate to be divided in justice to the widow,
son, and daughter?
82. My carpet is 22 feet across. My stride, either backwards or
forwards, is always 2 feet, and I make a stride every second. If I take
three strides forwards and two backwards continuously until I cross the
carpet, how long does it take me to reach the end of it?
NO TIME TO BE LOST
83. A merchant at Lisbon has an urgent business call to New York. Taking
these places to be, as they appear on a map of the world, on the same
parallel of latitude, and at a distance measured along the parallel, of
some 3600 miles, if the captain of a vessel chartered to go there sails
along this parallel, will he be doing the best that he can for the
impatient merchant?
ROUND THE ANGLES
84. Two schoolboys, John and Harry, start from the right angle of a
triangular field, and run along its sides. John’s speed is to Harry’s as
13 is to 11.
They meet first in the middle of the opposite side, and again 32 yards
from their starting point. How far was it round the field?
AN ECHO FROM THE PAST
85. The following question is given and spelt exactly as it was
contributed to a puzzle column by “John Hill, Gent.,” in 1760:--
“A vintner has 2 sorts of wine, viz. A and B, which if mixed in equal
parts a flagon of mixed will cost 15 pence; but if they be mixed in a
_sesqui-alter_ proportion, as you should take two flagons of A as often
as you take three of B, a flagon will cost 14 pence. Required the price
of each wine singly.”
PROMISCUOUS CHARITY
86. A man met a beggar and gave him half the money he had in his pocket,
and a shilling besides. Meeting another he gave him half of what was
left and two shillings, and to a third, he gave half of the remainder
and three shillings. This left a shilling in his pocket. How much had he
at first?
87. A young clerk wishes to start work at an office in the City on
January 1st. He has two promising offers, one from _A_ of £100 a year,
with a yearly rise of £20, the other from _B_ of £100 a year with a
half-yearly rise of £5. Which should he accept, and why?
HOW CAN I PAY MY BILL?
88. I have an abundance of florins and half-crowns, but no other coins.
In how many different ways can I pay my tailor £11, 10s. without
receiving change?
89. A monkey climbing up a greased pole ascends 3 feet and slips down 2
feet in alternate seconds till he reaches the top. If the pole is 60
feet high, how long does it take him to arrive there?
“SAFE BIND, SAFE FIND”
90. Old Adze, the village carpenter, who kept his tools in an open
chest, found that his neighbours sometimes borrowed and forgot to return
them.
To guard against this, he secured the lid of the chest with a letter
lock, which carried six revolving rings, each engraved with twelve
different letters. What are the chances against any one discovering the
secret word formed by a letter on each ring, which will open the lock,
and be the only key to the puzzle?
91. Five merry married couples happened to meet at a Swiss hotel, and
one of the husbands laughingly proposed that they should dine together
at a round table, with the ladies always in the same places so long as
the men could seat themselves each between two ladies, but never next to
his own wife. How long would their nights at the round table be
continued under these conditions?
SOUNDING THE DEPTH
92. In calm water the tip of a stiff rush is 9 inches above the surface
of a lake. As a steady wind rises it is gradually blown aslant, until at
the distance of a yard it is submerged. Can you decide from these data
the depth of the water in which the rush grows?
AMINTA’S AGE
93.
If to Aminta’s age exact
Its square you add, and eighteen more,
And from her age its third subtract,
And to the difference add three score,
The latter to the former then
Will just the same proportion bear
As eighteen does to nine times ten.
Can you Aminta’s age declare?
ONE FOR THE PARROT
94. A bag of nuts was to be divided thus among four boys:--Dick took a
quarter, and finding that there was one over when he made the division,
gave it to the parrot. Tom dealt in exactly the same way with the
remainder, as did Jack and Harry in their turns, each finding one nut
from the reduced shares to spare for the parrot. The final remainder
was equally divided among the boys, and again there was one for the
bird. How many nuts, at the lowest estimate, did the bag contain?
95. Here is an easy one:--
If five times four are thirty-three,
What will the fourth of twenty be?
96. What fraction of a pound, added to the same fraction of a shilling,
and the same fraction of a penny, will make up exactly one pound?
MENTAL ARITHMETIC
97. “Now, boys!” said Dr Tripos, “I think of a number, add 3, divide by
2, add 8, multiply by 2, subtract 2, and thus arrive at twice the number
I thought of.” What was it?
98. Two club friends, _A_ and _B_, deposit similar stakes with _C_, and
agree that whoever first wins three games at billiards shall take the
whole of them. _A_ wins two games and _B_ wins one. Upon this they
determine to divide the stakes in proper shares. How must this division
be arranged?
99. Not so simple as it sounds is the following compact little
problem:--If I run by motor from London to Brighton at 10 miles an hour,
and return over the same course at 15 miles an hour, what is my average
speed?
100. “I can divide my sheep,” says Farmer Hodge, who from his schooldays
had a turn for figures, “into two unequal parts, so that the larger part
added to the square of the smaller part shall be equal to the smaller
part added to the square of the larger part.”
How many sheep had the farmer?
A HELPFUL BURDEN
101. The following question was proposed in an old book of Mathematical
Curiosities published more than a hundred years ago:--
“It often happens that if we take two horses, in every respect alike,
yet, if both are put to the draught, that horse which is most loaded
shall be capable of performing most work; so that the horse which
carries the heavier weight can draw the larger load. How is this?”
102. In the king’s treasury were six chests. Two held sovereigns, two
shillings, and two pence, in equal numbers of these coins. “Pay my
guard,” said the king, “giving an equal share to each man, and three
shares to the captain; give change if necessary.” “It may not be
possible,” replied the treasurer, “and the captain may claim four
shares.” “Tut, tut,” said the king, “it can be done whatever the amount
of the treasure, and whether the captain has three shares or four.”
Was the king right? If so, how many men were there in the guard?
NUTS TO CRACK
103. I bought a parcel of nuts at forty-nine for twopence. I divided it
into two equal parts, one of which I sold at twenty-four, the other at
twenty-five for a penny. I spent and received an integral number of
pence, but bought the least possible number of nuts. How many did I buy?
What did they cost? What did I gain?
MONEY MATTERS
104. My purse contained sovereigns and shillings. After I had spent half
of its contents there were as many pounds left as I had shillings at
first. With what sum did I start?
A DELICATE QUESTION
105. A lady was asked her age in a letter, and she replied by postcard
thus:--
If first my age is multiplied by three,
And then of that two-sevenths tripled be,
The square root of two-ninths of this is four,
Now tell my age, or never see me more!
What was her age?
106. If cars run at uniform speed on the twopenny tube, from Shepherd’s
Bush to the Bank at intervals of two minutes, how many shall I meet in
half an hour if I am travelling from the Bank to Shepherd’s Bush?
PAYMENT BY RESULTS
107. What would it cost me to keep my word if I were to offer my
greengrocer a farthing for every different group of ten apples he could
select from a basket of a hundred apples?
108. If the minute-hand of a clock moves round _in the opposite
direction_ to the hour-hand, what will be the real time between three
and four, when the hands are exactly together?
109. Two monkeys have stolen some filberts and some walnuts. As they
begin their feast they see the owner of the garden coming with a stick.
It will take him two and a half minutes to reach them. There are twice
as many filberts as walnuts, and one monkey finishes the walnuts at the
rate of fifteen a minute in four-fifths of the time and bolts. The other
manages to finish the filberts just in time.
If the walnut monkey had stopped to help him till all was finished, when
would they have got away if they ate filberts at equal rates?
110. A cashier, in payment for a cheque, gives by mistake pounds for
shillings and shillings for pounds. The receiver spends half-a-crown,
and then finds that he has twice as much as the cheque was worth. What
was its value?
111. What five uneven figures can be added together so as to make up 14?
112. Three posts which vary in value are vacant in an office. In how
many ways can the manager fill these up from seven clerks who apply for
the appointments?
113. “It is now ⁵⁄₁₁ of the time to midnight,” said the fasting man, who
began his task at noon. What time was it?
A STRIKING TIME
114. If a clock takes six seconds to strike six how long will it take to
strike eleven?
WHAT ARE THE ODDS?
115. How would you arrange twenty horses in three stalls so as to have
an odd number of horses in each stall?
116. Here is a pretty little problem, which has at any rate an Algebraic
form, and is exceedingly ingenious:--
Given _a_, _b_, _c_, to find _q_.
WHEN WAS HE BORN?
117. Tom Evergreen was asked his age by some men at his club on his
birthday in 1875. “The number of months,” he said, “that I have lived
are exactly half as many as the number which denotes the year in which I
was born.” How old was he?
118. Draw three circles of any size, and in any position, so long as
they do not intersect, or lie one within another. How many different
circles can be drawn touching all the three?
119. We have seen that the nine digits can be so dealt with, using each
once, as to add up to 100. How can 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 be
arranged so that they form a sum which is equal to 1?
120. How is it possible, by quite a simple method, to find the sum of
the first fifty numbers without actually adding them together?
121. Two tram-cars, _A_ and _B_, start at the same time. _A_ runs into a
“lie by” in four minutes, and waits there five minutes, when _B_ meets
and passes it. Both complete the whole course at the same moment. In
what time can _A_ complete it without a rest?
A CURIOUS DEDUCTION
122. Take 10, double it, deduct 10, and tell me what remains.
123. The average weight of the Oxford crew is increased by 2 lbs., when
one of them, who weighs 12 stone, is replaced by a fresh man. What is
the fresh man’s weight?
ASK ANY MOTORIST
124. A motor car is twice as old as its tyres were when it was as old as
its tyres are. When these tyres are as old as the car is now, the
united ages of car and tyres will be two years and a quarter. What are
their respective ages now?
125. _A_ and _B_ on the edge of a desert can each carry provisions for
himself for twelve days. How far into the desert can an advance be made,
so that neither of them misses a day’s food?
126. A bottle of medicine and its cork cost half-a-crown, but the bottle
and the medicine cost two shillings and a penny more than the cork. What
did the cork cost?
IN A FIX
127. A boat’s crew are afloat far from land with no sail or oars. How
can they, without making any use of wind or stream, and without any
outside help, regain the shore by means of a coil of rope which happens
to be at hand.
128. What is the largest sum in silver that I can have in my pockets
without being able to give change for a half-sovereign.
129. I have apples in a basket. Without cutting an apple I give half of
the number and half an apple to one person; half of what then remains
and half an apple to another, and half of what are still left and half
an apple to a third. One apple now remains in the basket. How many were
there at first?
A QUEER DIVISION
130. A third of twelve divide
By just a fifth of seven;
And you will soon decide
That this must give eleven.
131. A motor goes 9 miles an hour uphill, 18 miles an hour downhill, and
12 miles an hour on the level. How long will it take to run 50 miles and
return at once over the same course?
132. In firing at a mark _A_ hits it in two out of three shots, _B_ in
three out of four, and _C_ in four out of five. The mark was hit 931
times. If each fired the same number of shots, how many hits did each
make, and how many shots were fired?
133. If a cat and a dog, evenly matched in speed, run a race out and
back over a course 75 yards in all, and the dog always takes 5 feet at a
bound and the cat 3 feet, which will win?
IN A FOG
134. In a fog a man caught up a wagon going in the same direction at 3
miles an hour. If the wagon was just visible to him at a distance of 55
yards, and he could see it for five minutes, at how many miles an hour
was he walking?
135. Three horses start in a race. In how many different ways can they
be placed by the judge?
NEW ZEALAND FOOTBALL
136. The New Zealanders, winning a match against Oxbridge, scored 34
points, from tries and tries converted into goals.
If every try had been converted they would have made four-fifths of the
maximum which a score of 34 points from tries and goals can yield.
What is this maximum, and what was their actual score?
137. What is the smallest number, of which the alternate figures are
cyphers, that is divisible by 9 and by 11?
138. Here is an interesting little problem:--_A_, with 8d. in his hand,
meets _B_ and _C_, who have five and three loaves respectively. In
hungry mood they all agree to share the loaves equally, and to divide
the money fairly between _B_ and _C_. How much does each receive?
THE MONEY-BOXES
139. When four money-boxes, containing pennies only, were opened and
counted, it was found that the number in the first with half those in
all the others, in the second with a third of all the others, in the
third with a fourth of all the others, and in the fourth with a fifth of
all the others, amounted in each case to 740. How much money did the
boxes contain, and how was it divided?
140. Two steamers start together to make a trip to a far-off buoy and
back. Steamer _A_ runs all the time at 10 miles an hour. Steamer _B_
does the passage out at 8 miles an hour, and the return at 12 miles.
Will they regain port together?
141. A golf player has two clubs mended in London. One has a new head,
the other a new leather face. The head costs four times as much as the
face. At St Andrews it costs five times as much, and the leather face at
St Andrews is half the London price. Including a shilling for a ball he
pays twice the St Andrews charges for these repairs. What is the London
charge for each?
FROM TWO WRONGS TO MAKE A RIGHT
142. Two children were asked to give the total number of animals in a
pasture, where sheep and cattle were grazing. They were told the numbers
of sheep and of cattle, but one subtracted, and gave 100 as the answer,
and the other arrived at 11,900 by multiplication. What was the correct
total?
THE STONE CARRIER
143. Fifty-two stones are placed at intervals along a straight road. The
distance between the first and the second is 1 yard, between the second
and the third it is 3 yards, between the next two 5 yards, and so on,
the intervals increasing each time by 2 yards.
How far would a tramp have to travel to earn five shillings promised to
him when he had brought them one by one to a basket placed at the first
stone?
144. On a division in the House of Commons, if the Ayes had been
increased by fifty from the Noes, the motion would have been carried by
five to three. If the Noes had received sixty votes from the Ayes, it
would have been lost by four to three. Did the motion succeed? How many
voted?
A WATCH PUZZLE
145. How many positions are there on the face of a watch in which the
places of the hour and minute-hands can be interchanged so as still to
indicate a possible time?
CRICKET SCORES
146. In a cricket match the scores in each successive innings are a
quarter less than in the preceding innings. The match was played out,
and the side that went in first won by fifty runs. What was the complete
score?
HOW HIGH CAN YOU THROW?
147. A boy throws a cricket ball vertically upwards, and catches it as
it falls just five seconds later. How high from his hands does the ball
go?
MEASURE THE CARPET
148. It may be said of a section of the gigantic carpet at Olympia that
had it been 5 feet broader and 4 feet longer it would have contained 116
more feet; but if 4 feet broader and 5 longer the increase would have
been but 113 feet. What were its actual breadth and length?
149. In estimating the cost of a hundred similar articles, the mistake
was made of reading pounds for shillings and shillings for pence in each
case, and under these conditions the estimated cost was £212 18s. 4d. in
excess of the real cost. What was the true cost of the articles?
SQUARES IN STREETS
150. The square of the number of my house is equal to the difference of
the squares of the numbers of my next door neighbour’s houses on either
side.
My brother, who lives in the next street, can say the same of the number
of his house, though his number is not the same as mine. How are our
houses numbered?
151. Two men of unequal strength are set to move a block of marble which
weighs 270 lbs., using for the purpose a light plank 6 feet long. The
stronger man can carry 180 lbs. How must the block be placed so as to
allow him just that share of the weight?
152. A man has twenty coins, of which some are shillings and the rest
half-crowns. If he were to change the half-crowns for sixpences and the
shillings for pence he would have 156 coins. How many shillings has he?
COUNT THE COINS
153. Some coins are placed at equal distances apart on a table, so that
they form the sides of an equilateral triangle.
From the middle of each side as many are then taken as equal the square
root of the number on that side, and placed on the opposite corner coin.
The number of coins on each side is then to the original number as 5 is
to 4. How many coins are there in all?
PUTTING IN THE POSTS
154. A gardener, wishing to fence round a piece of ground with some
light posts, found that if he set them a foot apart there would be 150
too few, but if placed a yard apart there would be 70 to spare. How many
posts had he?
155. _A_ gives _B_ £100 to buy 100 animals, which must be cows at £5
each, sheep at £1, and geese at 1s. How many of each sort can he buy?
156. John is twice as old as Mary was when he was as old as Mary is.
John is now twenty-one. How old is Mary?
157. In a cricket match _A_ made 35 runs; _C_ and _D_ made respectively
half and one-third as many as _B_, and _B_’s score was as much below
_A_’s as _C_’s was above _D_’s. What did _B_, _C_, and _D_ each score?
SETTLED BY REMAINDERS
158. What is the least number which, divided by 2, 3, 4, 5, 6, 7, 8, 9,
or 10, leaves respectively as remainders 1, 2, 3, 4, 5, 6, 7, 8, and 9?
TABLE TURNING
159. A square table stands on four legs, which are set at the middle
point of its sides. What is the greatest weight that this table can
uphold upon one of its corners?
BRIBING THE BOYS
160. Well pleased with the inspector’s report, the rector of a country
parish came into his school with 99 new pennies in his pocket, and said
that he would give them to the five boys in Standard VII. if they could,
within an hour, show him how to divide them so that the first share
should exceed the second by 3, be less than the third by 10, be greater
than the fourth by 9, and less than the fifth by 16. What was the answer
which would satisfy these conditions?
SHEEP-STEALING
161. Some Indian raiders carried off a third of a flock of sheep, and a
third of a sheep. Another party took a fourth of what remained, and a
fourth of a sheep. Others took a fifth of the rest and three-fifths of a
sheep. What was the number of the full flock, if there were then 409
left?
162. Three boys begin to fill a cistern. _A_ brings a pint at the end of
every three minutes, _B_ a quart every five minutes, and _C_ a gallon
every seven minutes. If the cistern holds fifty-three gallons, in what
time will it be filled, and who will pour in the last contribution?
ESTIMATE OF AGE
163. A man late in the last century said that his age was the square
root of the year in which he was born. In what year did he say this?
A DEAL IN CHINA
164. A dealer in Eastern curios sold a Satzuma vase for £119, and on
calculation found that the number which expressed his profit per cent.
expressed also the cost price in pounds of the vase. What was this
number?
165. What is the chance of throwing at least one ace in a single throw
with a pair of dice?
166. A thief starts running from a country house as fast as he can. Four
minutes later a policeman starts in pursuit. If both run straight along
the road, and the policeman gets over the ground one-third faster than
the thief, how soon will he catch him?
167. Twenty-seven articles are exposed for sale on one of the stalls of
a bazaar. What choice has a purchaser?
VERY PERSONAL
168. “How old are _you_, dad?” said Nellie on her birthday, as her
father gave her as many shillings as she was years old. His answer was
quite a puzzle for a time, but with the help of her schoolfellows Nellie
worked it out.
This is what he said:--
“I was twice as old as you are
The day that you were born.
You will be just what I was then
When fourteen years are gone.”
How old was Nellie, and how old was her dad?
WORD AND LETTER PUZZLES
A MAGIC COCOON
This Magic Cocoon is so cleverly spun that the word can be traced and
read in many ways.
N
N O N
N O O O N
N O O C O O N
N O O C O C O O N
N O O C O C O C O O N
N O O C O C O O N
N O O C O O N
N O O O N
N O N
N
How many readings can you discover starting from one or other of the Cs,
and passing up and down or sideways, but not diagonally, and never over
the same letter twice in a reading? There are 756!
THE CHRONOGRAM
The Chronogram, severely classed by Addison as “a species of false wit”
is a sentence in which the salient letters represent in Roman numerals
some particular year. A good English specimen is this: “My Day Closed Is
In Immortality.” The capital letters in these words give MDCIII., or
1603, the year in which Queen Elizabeth died.
A FRENCH CHRONOGRAM
The battle-cry at Montlhéry in 1465 was:--“à CheVaL, à CheVaL,
gendarMes, à CheVaL!” Taking the letters printed in capitals--
M = 1000
CCC = 300
LLL = 150
VVV = 15
----
we have the date of the battle 1465
A NOTABLE CHRONOGRAM
QVI CHRISTI IAVDES CANTANT
SANCTÆ PASSIONIS SVÆ VIRTVTE
IN IPSO ET PATRE VNVM SINT.
This curious inscription is placed over the organ at Ober Ammergau. Add
together its Roman numerals and they give the date at which the organ
was dedicated. The English of it is:--
May those who sing the praises of Christ be by virtue of His Sacred
Passion one in The Father and in Himself.
A PERFECT CHRONOGRAM
On a damaged inscription to Bishop Berkeley in Winchester Cathedral are
the words--VIXI, LVXI--I have lived, I have shone. Added together in
their values as Roman numerals the letters of these two words give his
age exactly at his death--eighty-three.
A PRIZE MOTTO
On the return of the C.I.V. from the Boer War a prize was offered by
_Truth_ for the best motto appropriate to them. This was to consist of
three words of which the first must begin with C the second with I and
the third with V.
The prize was taken by the following Latin motto which is singularly
happy both in construction and in meaning:--
CIVI IVI VICI
_I roused_ _I went_ _I won._
The sequence of events is perfect; no letters but C.I.V. are used and
the motto is a palindrome if read by syllables.
THE MUSICAL SCALE
As in olden days some of the Psalms and other writings were constructed
in acrostic form, so in the Middle Ages even serious writers would
juggle with letters, as though they felt that such tricky methods were
an aid to memory.
It was in this spirit that Guido Aretino, a Benedictine monk of Tuscany
in 1204, gave names to the notes used in the musical scale from the
first syllables of the lines of a Latin hymn. “Ut” is still used in
France, though we and the Italians have substituted “do.”
UT queant laxis REsonare fibris,
MIra gestorum FAmuli tuorum,
SOLve polutis LAbii reatis
O Pater alme!
ALL THE ALPHABET!
Many of us know that there is a long verse in the Book of Ezra in which
all the letters of the alphabet are used, taking “j” as “i” (_Ezra_
vii., v. 21).
This very curious coincidence also occurs in a comparatively short
sentence in “The Beth Book,” by Sarah Grand:--“It was an exquisitely
deep blue just then, with filmy white clouds drawn up over it like
gauze;” and here “j” is itself in evidence.
APT ALLITERATION
Schopenhauer, the famous German philosopher, who was a confirmed
bachelor and misogynist, was compelled while living at Frankfort to
support an old lady who had been crippled by his violence. When her
death came as a welcome relief to him, he composed the following clever
epitaph:--
Obit anus,
Abit onus.
which by the interchange of two letters pictured the position. It may be
freely rendered:--
Old lady dies,
My burden flies.
A DOUBLE SEQUENCE
The following clever composition, which appeared in the pages of
_Truth_, contains a _double_ sequence of words, which increase a letter
at a time, the same letters appearing in varied order until at last “o”
culminates in _thornless_, and “a” in _restrainest_. It is quite a
remarkable _tour-de-force_.
O lack-_a_-day! _at_ eve we _sat_,
One _star_ had lit its lamps _on_ high.
We did _not note_ the circling bat,
_Start_ from the _stone_ when flitting nigh.
For the _strait_ gate of _honest_ doubt
Shut off the _thrones_ of Love and Gain;
We dreamed not, as we mourned without,
That Time’s swift _transit shortens_ pain.
O Thou, Who _trainest_ souls to shine,
Though once we craved a _thornless_ lot,
This gracious truth we now divine:
The bruised reed Thou _strainest_ not;
But by _restraints_, that gently tame;
_Restrainest_ Passion’s kindling flame.
THE REIGN OF TERROR
During the Reign of Terror, France and her people and position were thus
alphabetically described:--
Le peuple Français A B C. (abaissé).
La gloire nationale F A C. (effacée).
Les places fortes O Q P. (occupées).
Quarante trois députés C D. (cédés).
L’armée D P C. (dépaysée).
Les ministres A J. (agés).
La liberté O T. (ôtée).
La charte L U D. (éludée).
SEE-SAW
This elaborate method of piling up no less than seven consecutive
“thats,” so that they make tolerable sense, was told to his boys during
school-time by Dr Moberly, then headmaster of Winchester, and afterwards
Bishop of Salisbury, just fifty years ago:--
I saw that C saw.
C saw that that I saw.
I saw that that that C saw was so.
C saw that, that that that I saw was so.
I saw that, that _that_ that that C saw was so.
C saw _that_ that, that _that_ that that I saw was so.
I saw _that_ that, that _that_ that that _that_ C saw was so.
FOR A ROMAN HOLIDAY
If the Roman ladies and children, at their equivalent for Christmas,
amused themselves by acting verbal charades, an excellent word was at
their disposal, “sustineamus”--“let us endure,” which can be broken up
exactly into _sus_, _tinea_, _mus_--_a sow_, _a moth_, _a mouse_.
A WORD SQUARE
1. Can you complete this word square, so that its four words read alike
from top to bottom and from left to right?
* E * *
E * * *
* * * E
* * E *
ANOTHER WORD SQUARE
2. Can you fill in this word square?
C * * C * E
* N U * E S
* U * E S *
C * E * S *
* E S * * E
E * T E * *
DUPLICATE LETTERS
3. In this sentence, when complete,
So****AG****LATI****X****ITH
each group of four missing letters contains two pairs of letters which
are alike. Can you on these lines complete the sentence?
Here is a similar sentence by way of illustration:
T****M****TERTAIN****MUND,
which becomes when filled in--
T_wo_ _wo_m_en_ _en_tertain_ed_ _Ed_mund.
ANOTHER WORD SQUARE
4. Can you complete this word square by substituting letters for the
dots?
W * * * E
* * T * *
* T O N *
* * N * *
E * * * T
WORD BUILDING
5. What word can be made with these?
L S D U D O D U D.
6. A lovelorn youth consulted a married lady on his condition, and was
asked by her on a slip of paper:--
“Loruve?”
When he had deciphered this, and had answered in the affirmative, she
handed to him another slip, on which this advice was written:--
L
“Prove A F and ensure success.”
D
What did it all mean?
A DOUBLE ACROSTIC
7. _Saint of Spain, whose daily word
Twenty years hath London heard!_
Sweet days, elastic metal, motion sharp.
My halls once echoed to an Irish harp.
The “son of sorrow,” honourable of yore.
Dread goddess, loosing the loud dogs of war.
Time’s atom--total of eternities.
This name an insect bears, a patriot bore.
“So do, yet hear me,” said Themistocles.
ANOTHER WORD SQUARE
8. Can you complete this word square?
* M * N * S
M * N * O *
* N * B * E
N * B * L *
* O * L * R
S * E * R *
A FRENCH ORACLE
9. A spruce young Frenchman at a _fête_ consulted a modern oracle as to
how he could best please the ladies. This was the mystic response:--
MEC DO BIC.
Can you interpret it?
A QUAINT EQUATION
10. In our young days we have often wrestled with vulgar fractions, but
apart from Algebra we have had no serious concern with any in which
letters take the place of figures. A specimen of this sort, not known to
science, is the following curiosity:--
m
--
ot
-- = mo.
y
11. The puzzle in _Truth_ was recently founded upon “ourang-outang,”
which had been cleverly buried. We will give a few of the best results.
This is one:--
Poor wretch! a moisture filled his eye,
“Do not rebuff a lonely boy,”
Said he, “If ere I sink and die
Your smile--O! pardon will be joy!”
Another is:--
Though I jump, and row, and run,
Cap or cup I never won.
What animals are buried in these lines?
LIKE A PEACOCK’S TAIL
12. Fourteen letters here we fix,
Vowels only two are spoken;
All together these we mix
Into what can not be broken.
A WEIRD WORD
13. There is an English word of thirteen letters in which the same vowel
occurs four times, the same consonant six times, another consonant
twice, and another once. Can you hit upon it?
A CONDENSED PROVERB
14. Though brevity is said to be the soul of wit, we are too often
flooded nowadays with a superabundance of words.
Here is an attempt at modest condensation. A familiar English proverb is
quite clearly expressed to the solver’s seeing eyes in this brief
phrase:--
WE IS DO
What is the proverb?
ANOTHER WORD SQUARE
15. Can you complete this word square?
W * * * * S
* R * * R *
* * O R * *
* * R M * *
* R * * N *
S * * * * M
CAN YOU DECIPHER IT?
16. The following puzzle lines are attributed to Dr Whewell:--
O O N O O.
-----------
U O A O O I O U
O N O O O O M E T O O.
U O A O I D O S O
I O N O O I O U T O O!
A BROKEN DIAMOND
17. Can you fill in the vacancies in this diamond?
P
F O *
C * R * *
F * * C * * *
P O R C E L A I N
R * * L * * *
S * A * *
S I *
N
Its words must read alike from left to right and from top to bottom.
WHAT IS THIS?
18. Tan HE Edsa VEN in
It N Gja SmeTs AsgN
aD Az Rett De.
A PHONETIC JOURNEY
19. I can travel first-class on the Great Eastern Railway from 2 2 2 2 2
2 2 2 4 4 4 4 4 5 0 0. What is the cost of my journey, and its length in
time?
A CURIOUS OLD INSCRIPTION
20. Seogeh sreve ereh wcisume vahl
Lah sehs se otreh nos llebdnas
Regni freh nos gnires rohyer
Ganoed iryd ale nifae esots sorcy
Rub nabot es rohk co caed ir.
Can you decipher it?
IRISH STEW AT SIMPSON’S
21. I wrote the following note recently:--
Dear Jack,--Meet me at Simpson’s to-morrow at 1.30. We will sample
their excellent Irish stew. Here are some catchwords that will remind
you of the invitation:--
Join me at and
i
s
Why should they remind him of it?
ENGLISH AS SHE IS SPELT!
22. This was the exact text of a letter sent to the master of an English
village school by a labourer as an excuse for his boy’s absence:--
“Cepatomtogoatatrin”
Can you decipher it?
A DOUBLE ACROSTIC
23. This double Acrostic will afford an easy exercise in mental
gymnastics for those to whom such pastime appeals:--
Now we are fain
To rack your brain.
1. More fit for babes and sucklings than for you.
2. Robbed of externals this is very true.
3. Diminutive in measure and in weight.
4. Pen-name of one a true pen potentate.
5. A palindrome quite plain is here in sight.
6. Sans head and tail it also yields this light.
7. Here is in short what anyone may write.
FIND THE PROVERB
24. c e f h i m n o r s t v y
3 2 2 2 7 1 1 2 6 5 8 3 9
4 4 1 2 1 6 1
3 1 5 8 2 3
5 7 9 6
4
The letters with ones under them are the first letters of words, those
with twos under them are second letters of words, and so on.
A PUZZLE WILL
25. Having occasion to make a few slight additions to my will, I called
in my lawyer to arrange the matter. How far forward did the instructions
contained in the following lines carry him in his work?
Set down a hundred in my will,
Add nothing to the text;
Five hundred now a space may fill,
And one be added next.
Another hundred write as well,
And yet another one;
Then fifty more, and try to tell
The deed that now is done.
ANOTHER WORD SQUARE
26. Can you complete this word square?
* D * * O *
D * * I * E
* S * A * D
T * A * A *
* R * A * E
R * D * E *
MUSICAL EPITAPHS
27. Over the grave of a French musician, who was choked by a fish bone,
the following epitaph was inscribed in notes of music:--A. G. A. E. A.
Over the porch of the house of Gustave Doré these musical notes were
placed on a tablet:--C. E. B. A. C. D.
What do these inscriptions signify?
QUITE TOO TOO
28. “Where can we meet to-morrow?” said Jack Spooner to his best girl.
“We will go,” she replied, “at 222222222222 LEY STREET.”
When and where did they meet?
A BROKEN WORD
29. What does this spell?
C
-------------------
T T T T T T T T T T
CONTRADICTORY TERMS
30. What English word is it which may be so treated as to affirm or
disallow the use of its own initial or final letter?
PRINTERS’ PIE
31. Can you arrange these letters
E I O O O U
B C N N R R S S
so that they form the title of a book well-known to boys?
FILL IN THE GAPS
32. Keeping these letters in their present order make a sensible
sentence by inserting among them as often as is necessary another
letter, which must be in every case the same.
A DEN I I CAN DOCK.
DISTORTED SHAKESPEARE