If the number 37 be multiplied by 3, or any multiple of 3 up to 27,
the product is expressed by three similar digits. Thus--
37 × 3 = 111
37 × 6 = 222
37 × 9 = 333
The products succeed each other in the order of the digits read
downwards, 1, 2, 3, etc., these being multiplied by 3 (their number
of places) reproduce the multiplicand of 37.
1 × 3 = 3
2 × 3 = 6
3 × 3 = 9
If it be multiplied by multiples of 3, beyond 27, this peculiarity is
continued, except that the extreme figures taken together represent
the multiple of 3 that is used as a multiplier. Thus--
37 × 30 = 1110, extreme figures, 10
37 × 33 = 1221 " " 11
37 × 36 = 1332 " " 12
The number 73 (which is 37 inverted) multiplied by each of the
numbers of arithmetical progression 3, 6, 9, 12, 15, etc., produces
products terminating (unit’s place) by one of the ten different
figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These figures will be found in
the reverse order to that of the progression, 73 × 3 produces 9, by 6
produces 8, and 9 produces 7, and so on.
Another number which falls under some mysterious law of series is
142,857, which, multiplied by 1, 2, 3, 4, 5, or 6 gives the same
figures in the same order, beginning differently; but if multiplied
by 7, gives all 9’s.
142,857 multiplied by 1 = 142,857
" " 2 = 285,714
" " 3 = 428,571
" " 4 = 571,428
" " 5 = 714,285
" " 6 = 857,142
" " 7 = 999,999
Multiplied by 8, it gives 1,142,856, the first figure added to the
last makes the original number--142,857.
The vulgar fraction 1/7 = ·142,857.
The following number, 526315789473684210, if multiplied as above,
will, in the product, present the same peculiarities, as also will
the number 3448275862068965517241379310.
The multiplication of 987654321 by 45 = 444444444445
Do. 123456789 " 45 = 5555555505
Do. 987654321 " 54 = 53333333334
Do. 123456789 " 54 = 6666666606
Taking the same multiplicand and multiplying by 27 (half 54) the
product is 26,666,666,667, all 6’s except the extremes, which read
the original multiplier (27). If 72 be used as a multiplier, a
similar series of progression is produced.
6. In stables five, can you contrive to put in horses twenty--
In each stable an odd horse, and not a stable empty?
“THREE THREES ARE TEN.”