THE ANALYSIS OF CHANCE AND MATHEMATICAL GAMES.
MAGIC SQUARES—THE SIXTEEN PUZZLE—SOLITAIRE—EQUIVALENTS.
We will now proceed to draw our readers’ attention to several
experiments very famous at a former period, but which our own
generation has completely overlooked. We refer to the Analysis of
Chance, a science still known under the title of _Calculation of
Probabilities_, formerly cultivated with so much ardour, but to-day
almost fallen into oblivion.
Originating in the caprice of the clever Chevalier de Méré, who in
1654 suggested the game to Pascal, the analysis of chance has given
rise to investigations of an entirely novel kind, and attempts have
been made to measure the mathematical degree of credence to be given
to simple conjectures. We will first recapitulate the principles laid
down by Laplace on this subject. We know that of a certain number of
events, one only can happen, but nothing leads us to the belief that
one will happen more than the other. The theory of chance consists in
reducing all the events of the same kind to a certain number of equally
possible cases, such, that is to say, that we are equally undecided
about, and to determine the number of cases favourable to the event,
whose probability we are seeking. The ratio of this number to that of
all possible cases is the measure of this probability, which is thus a
fraction, the numerator of which is the number of favourable cases, and
the denominator the number of all possible cases. When all the cases
are favourable to an event, its probability changes to certainty, and
it is then expressed by the unit. Probabilities increase or diminish
by their mutual combination; if the events are independent of each
other, the probability of the existence of their whole is the product
of their particular probabilities. Thus the probability of throwing an
ace with one dice being 1/6, that of throwing two aces with two dice is
1/36. Each of the sides of one dice combining with the six sides of the
other, there are thirty-six possible cases, among which one only gives
the two aces. When two events depend on each other, the probability
of the double event is the product of the probability of the first
event by the probability that, that event having occurred, the other
will occur. This rule helps us to study the influence of past events
on the probability of future events. If we calculate _á priori_ the
probability of the event that has occurred and an event composed of
this and another expected event, the second probability divided by the
first, will be the probability of the expected event, inferred from the
observed event.
The probability of events serves to determine the hope or fear of
persons interested in their existence. The word _hope_ here expresses
the advantage which someone expects in suppositions which are only
probable. This advantage in the theory of chances is the product
of the hoped-for sum by the probability of obtaining it; it is the
partial sum which should arise when one does not wish to run the
risks of the event, supposing that the apportionment corresponds
to the probabilities. This apportionment is only equitable when we
abstract from it all foreign circumstances; because an equal degree of
probability gives an equal title to the hoped-for sum. This advantage
is called _mathematical hope_. Nevertheless, the rigorous application
of this principle may lead to an inadmissible consequence. Let us see
what Laplace says. Paul plays at heads and tails, on the understanding
that he receives two shillings if he succeeds at the first throw,
four shillings if he succeeds at the second, eight at the third, and
so on. His stake on the game, according to calculation, must be equal
to the number of throws; so that if the game continues indefinitely,
the stake also continues indefinitely. Yet, no reasonable man would
venture on this game even a moderate sum, £2 for example. Whence,
therefore, comes this difference between the result of the calculation,
and the indication of common-sense? We soon perceive that it proceeds
from the fact, that the moral advantage which a benefit procures for
us is not proportional to this advantage, and that it depends on a
thousand circumstances, often very difficult to define, but the chief
and most important of which is chance. In fact, it is evident that a
shilling has much greater value for one who has but a hundred than
for a millionaire. We must, therefore, distinguish in the hoped-for
good between its absolute and its relative value; the latter regulates
itself according to the motives which cause it to be desired, while
the former is independent. In the absence of a general principle
to appreciate this relative value, we give a suggestion of Daniel
Bernouilli which has been generally admitted.
The relative value of an extremely small sum is equal to its absolute
value, divided by the total advantage of the interested person. On
applying the calculus to this principle, it will be found that the
_moral hope_, the growth of chance due to expectations, coincides with
the mathematical hope, when chance, considered as a unit, becomes
infinite in proportion to the variations it receives from expectations.
But when these variations are a sensible portion of the unit, the
two hopes may differ very greatly from each other. In the example
cited, this rule leads to results conformable to the indications of
common-sense. We find, in point of fact, that if Paul’s fortune amounts
only to £8, he cannot reasonably stake more than 7_s._ on the game.
At the most equal game, the loss is always, relatively greater than
the gain. Supposing, for example, that a person possessing a sum of
£4, stakes £2 on a game of heads or tails, his money after placing
his stake will be _morally_ reduced to £3 11_s._ 0_d._—that is to
say, _this latter sum will procure him the same moral advantage as
the condition of his funds after his stake_. Whence we draw this
conclusion: that the game is disadvantageous, even in the cases
where the stake is equal to the product of the sum hoped for by the
probability. We may, therefore, form an idea of the immorality of games
in which the hoped-for sum is below this product.
[Illustration: Fig. 857.—The game of the needle.]
Jacques Bernouilli has thus laid down the result of his investigations
on the calculation of probabilities. An urn containing white and
black balls is placed in front of the spectator, who draws out a
ball, ascertaining its colour, and puts it back in the urn. After a
sufficient number of draws, the total number of _extracted_ balls
divided by the total number of balls represents a fraction very near
to that which has for a numerator the real number of white balls
existing in the urn, and for the denominator the total number of balls.
In other words, the ratios of the number either of extracted white
balls, or the whole of the white balls to the total number, tend to
become equal; that is, the probability derived from this experiment
approaches indefinitely towards a certainty. The two fractions may
differ from each other as little as possible, if we increase the number
of draws. From this theorem we deduce several consequences.
1. The relations of natural effects are nearly constant when these
effects are considered in a great number.
2. In a series of events indefinitely prolonged, the action of regular
and constant causes affects that of irregular causes.
_Applications._—The combinations presented by these games have been the
subject of former researches regarding probabilities. We will complete
our exposition with two more examples.
Two persons, A and B, of equal skill, play together on the
understanding that whichever beats the other a certain number of
times, shall be considered to have won the game, and shall carry off
the stakes. After several throws the players agree to give up without
finishing the game; and the point then to be settled, is in what manner
the money is to be divided between them. This was one of the problems
laid before Pascal by the Chevalier de Méré. The shares of the two
players should be proportional to their respective probabilities of
winning the game. These probabilities depend on the number of points
which each player requires to reach the given number. A’s probabilities
are determined by starting with the smallest numbers, and observing
that the probability equals the unit, when player A does not lose a
point. Thus, supposing A loses but one point, his chance is 1·2, 3·4,
7·8, etc., according as B misses one, two, or three points. Supposing A
has missed two points, it will be found that his chance is as 1·4, 1·2,
11·6, etc., according as B has missed one, two, or three points, etc.
Or we may suppose that A misses three points, and so on.
We should note, _en passant_, that this solution has been modified
by Daniel Bernouilli, by the consideration of the respective fortune
of the players, from which he deduces the idea of moral hope. This
solution, famous in the history of science, bears the name of the
_Petersburgh problem_, because it was made known for the first time in
the “Memoires de l’Académie de Russie.”
We will now describe the _game of the needle_. It is a genuine
mathematical amusement, and its results, indicated by theory, are
certainly calculated to excite astonishment. The game of the needle is
an application of the different principles we have laid down.
If we trace on a sheet of paper a series of parallel and equi-distant
lines, AA1, BB1, CC1, DD1, and throw down on the paper at hazard a
perfectly cylindrical needle, _a b_, the length of which equals half
the distance between the parallel lines (figs. 858 and 859), we
shall discover this curious result. If we throw down the needle a
hundred times, it will come in contact with one of the parallel lines
a certain number of times. Dividing the number of attempts with the
number of successful throws, we obtain as a quotient a number which
approaches nearer the value of the ratio between the circumference
and the diameter in proportion as we multiply the number of attempts.
This ratio, according to the rules of geometry, is a fixed number,
the numerical value of which is 3·1415926. After a hundred throws
we generally find the exact value up to the two first figures: 3·1.
How can this unexpected result be explained? The application of the
calculus of probabilities gives the reason of it. The ratio between
the successful throws and the number of attempts, is the probability
of this successful throw. The calculation endeavours to estimate this
probability by enumerating the possible cases and the favourable
events. The enumeration of possible cases exacts the application of
the principle of compound probabilities. It will be easily seen that
it suffices to consider the chances of the needle falling between two
parallel lines, AA1 and BB1 (fig. 858), and then to consider what
occurs in the interval, _m n_, equal to the equi-distance. To obtain a
successful throw, it is necessary then:—
1. That the middle of the needle should fall between _m_ and _l_,
the centre of _m o_. 2. That the angle of the needle with _m o_ will
be smaller than the angle, _m c b_. The calculation of all these
probabilities and their combination by multiplication, according to
the rules of compound probabilities, gives as the final expression of
probability the number.
[Illustration: Fig. 858.—The needle game.]
[Illustration: Fig. 859.—The needle game.]
This curious example justifies the theorem of Bernouilli relating to
the multiplication of events; there is no limit to the approximation
of the result, when the attempts are sufficiently prolonged. When the
length of the needle is not exactly half the distance between the
parallel lines, the practical rule of the game is as follows: The ratio
between the number of throws and the number of successful attempts must
be multiplied by double the ratio between the length of the needle and
the distance between the parallel lines. In the case cited above, the
double of the latter ratio equals the unit. We will give an application
to this. A needle two inches long is thrown 10,000 times on a series of
parallel lines, two-and-a-half inches apart; the number of successful
throws has been found to equal 5000. We take the ratio 1090/5009, and
multiply it by the ratio 1000/636 and the product is 3·1421. The true
value is 3·1415. We have an approximation of 6/10000.
The dimensions indicated in this experiment are those which present in
a given number of attempts the most chances of obtaining the greatest
possible approximation. We will conclude these remarks on games by some
observations borrowed from Laplace.
The mind has its illusions like the sense of sight; and just as the
sense of touch corrects the latter, reflection and calculation correct
the former. The probability founded on an every-day experience, or
exaggerated by fear or hope, strikes us as a superior probability, but
is only a simple result of calculation.
In a long series of events of the same kind, the mere chances of
accident sometimes offer these curious veins of good or bad fortune,
which many persons do not hesitate to attribute to a kind of fatality.
It often happens in games which depend both on chance and the
cleverness of the players, that he who loses, overwhelmed with his want
of success, seeks to repair the evil by rash playing, which he would
avoid on another occasion; he thus aggravates his own misfortune and
prolongs it. It is then, however, that prudence becomes necessary, and
that it is desirable to remember that the moral disadvantage attaching
to unfavourable chances is increased also by the misfortune itself.[41]
Mathematical games, formerly so much studied, have recently obtained a
new addition in the form of an interesting game, known as the “Boss”
puzzle. It has been introduced from America, and consists of a square
box, in which are placed sixteen small wooden dice, each bearing a
number (fig. 860). No. 16 is taken away, and the others are placed
haphazard in the box, as shown in fig. 861. The point is then to move
the dice, one by one, into different positions, so that they are at
last arranged in their natural order, from one to fifteen; and this
must be accomplished by slipping them from square to square without
lifting them from the box. If the sixteenth dice is added, the game
may be varied, and we may seek another solution of the problem, by
arranging the numbers so that the sum of the horizontal, vertical, and
diagonal lines gives the number 34. In this form the puzzle is one of
the oldest known. It dates from the time of the primitive Egyptians,
and has often been investigated during the last few centuries,
belonging, as it does, to the category of famous _magic squares_, the
principles of which we will describe. The following is the definition
given by Ozanam, of the Academy of Sciences, at Paris, at the end of
the seventeenth century. The term _magic square_ is given to a square
divided by several small equal or broken squares, containing terms of
progression which are placed in such a manner that all those of one
row, either across, from top to bottom, or diagonally, make one and the
same sum when they are added, or give the same product when multiplied.
It is therefore evident from this definition, that there are two kinds
of magic squares, some formed by terms of arithmetical progression,
others by terms of geometrical progression. We must also distinguish
the equal from the unequal magic squares.
[Illustration: Fig. 860.—The sixteen puzzle.]
[Illustration: Fig. 861.—The numbers placed at hazard, and No. 16
removed.]
We give here several examples of magic squares with terms of
mathematical progression, among them the square of 34, giving one of
the solutions to the puzzle just described (fig. 862). We also give an
example of a magic square composed of terms of geometrical progression.
The double progression for examples 1, 2, 4, 8, 16, 32, 64, 128, 256,
as here arranged (fig. 863), forms such a square that the product
obtained by multiplying the three terms of one row, or one diagonal,
is 4,096, which is the cube of the mean term 16. The squares have been
termed _magic_, because, according to Ozanam, they were held in great
veneration by the Pythagoreans. In the time of alchemy and astrology,
certain magic squares were dedicated to the seven planets, and engraved
on a metal blade which sympathized with the planet. To give an idea of
the combinations to which the study of magic squares lends itself, it
is sufficient to add that mathematicians have written whole treatises
on the subject. Frénicle de Bessy, one of the most eminent calculators
of the seventeenth century, consecrated a part of his life to the study
of magic squares. He discovered new rules, and found out the means of
varying them in a multitude of ways. Thus for the magic square, the
root of which is 4, only sixteen different arrangements were known.
[Illustration: Fig. 862.—Examples of magic squares formed by terms of
arithmetical progression.]
[Illustration: Fig. 863.—Magic square formed by terms of geometrical
progression.]
Frénicle de Bessy found 880 new solutions. An important work from the
pen of this learned mathematician has been published under the title of
“Carrés ou Tables Magiques,” in the “Memoirs de l’Académie Royale des
Sciences,” from 1666-1699, vol. v. Amateurs, therefore, who are accused
of occupying themselves with a useless game, unworthy the attention of
serious minds, will do well to bear in mind the works of Frénicle, and
better still, to consult them.
We have so far considered only the first part of the puzzle. We may
now examine the problem to which specially it has given rise. We are
quite in accord with M. Piarron de Mondesir, who has been so good as to
enlighten us upon the subject, which is really much more difficult than
it appears.
A French paper once proposed to give a prize of 500 francs to any
individual who would solve the following problem:—
Throw the numbers out of the box, replace them at hazard, then in
arranging them place them in the following order (A fig. 864).
[Illustration: Fig. 864.—The Sixteen Puzzle.]
Now nobody solved this problem, because in nine cases out of ten it is
impossible to do so. The first twelve numbers will come correctly into
their places, and even 13 can be put in its place without much trouble;
but, instead of getting the last row right we shall find it will come
out like B, viz., 14, 15, 13, in the large majority of instances. So
any case can be solved in one of the two results given above, and we
can tell in advance, without displacing a number, in which way the
puzzle will eventuate.
[Illustration: Fig. 865.—Example 1.]
[Illustration: Fig. 866.—Example 2.]
Let us give this problem our attention for a few minutes, and we shall
not find it difficult.
Take the first example. We will throw the cubes out of the box and put
them back in the order shown in fig. 865.
We see now that 1 occupies the place of 11, 11 that of 7, 7 that of 8,
8 that of 6, 6 of 15, 15 of 1. This much is evident without any study.
We formulate these figures as follows, beginning with 1 and working
from figure to figure till we are led to 1 again, and so on.
_1st. Series._—1, 11, 7, 8, 6, 15, 1 (6) even.
Counting the number of different cubes we have 6; and we put (6) in a
parenthesis. We call the first series even because 6 is an even number.
We now establish, by the same formula, a second series commencing with
2, _and going back to it_, thus—
_2nd Series._—2, 4, 2 (2) even.
_3rd Series._—3, 5, 10, 12, 3 (4) even.
_4th Series._—9, 13, 14, 9 (3) uneven.
We have now four series, the total number of points equal 15, as there
ought to be, for one cube is absent.
Let us now take another example (_see_ fig. 866), and by working as
before we have four series again, viz:—
_1st Series._—1, 7, 1 (2) even.
_2nd Series._—2, 11, 3, 8, 4, 15, 2 (6) even.
_3rd Series._—5, 12, 13, 5 (3) uneven.
_4th Series._—9, 14, 10, 9 (3) uneven.
This gives us only 14 as a total, because 6 has not been touched at all.
* * * * *
And now for the rule, so that we may be able to ascertain in advance,
when we have established our series, whether we shall find our puzzle
right or wrong at the end. We must put aside all unplaced numbers and
take no notice of uneven series. Only the even series must be regarded.
Thus if we do not find 1, or if we find 2, 4, or 6, the problem will
come into A as a result. If we find 1, 3, 5, or 7, the case will
eventuate as in B (fig. 864). Let us apply the rule to the problems we
have worked, and then the reason will be apparent.
In the first we find _three even series_; the problem will then end as
in B diagram (fig. 864), for the number of _like_ series is _odd_.
In the second we find _two even series_ (pairs); we shall find our
problem work out as in diagram A (fig. 864), for the number of _like
series_ is _even_, one pair in each.
We are now in possession of a simple rule, both rapid and infallible,
and which will save considerable trouble, as we can always tell
beforehand how our puzzle will come out. Any one can test the
practicability of the rules for himself, but we may warn the reader
that he will never be able to verify every possible instance, for the
possible cases are represented by the following sum—
2×3×4×5×6×7×8×9×10×11×12×13×14×15.
That is to say, 1,307,674,368,000 in all.
SOLITAIRE.
This somewhat ancient amusement is well known, and the apparatus
consists of a board with holes to receive pegs or cups to receive the
balls, as in the illustrations (figs. 867 and 870.) The usual solitaire
board contains thirty-seven pegs or balls, but thirty-three can also be
played very well. Many scientific people have made quite a study of the
game, and have published papers on the subject. M. Piarron de Mondesir
has given two rules which will prove interesting.
The first is called that of equivalents, and supposes the game to be
played out to a conclusion; the second, called the ring-game, admits of
a calculation being made so that the prospects of success can be gauged
beforehand.
The method of play is familiar, so we need not detail it. It is simply
“taking” the balls by passing over them in a straight line. The method
of “equivalents” consists in replacing one ball with two others, as we
will proceed to explain by the diagram (fig. 868).
[Illustration: Fig. 867.—Solitaire.]
Suppose we try the 33 game, which consists in filling every hole with
the exception of the centre one, and in “taking” all the balls, leaving
one solitary in the centre at the last. Suppose an inexperienced player
arrives at an impossible solution of five balls in 4, 11, 15, 28, and
30.
To render the problem soluble, and to win his game, I will replace No.
11 by two equivalents, 9 and 10, the ball 28 by two others, 23 and 16,
and the ball 30 by 25 and 18. These substitutions will not change the
“taking off,” for I can take 10 with 9, 23 with 16, and 25 with 18. But
by so doing I substitute for an irreducible solution of five balls a
new system of eight (those shown with the line drawn through them in
the diagram), which can easily be reduced to the desired conclusion,
and the game will be achieved.
There are in reality three terminations possible to the problem—the
single ball, the _couple_, and the _tierce_; that is, you may have only
one left, or two placed diagonally, such as 9-17, 25-29, or a system of
three in a straight line, 9-16-23. By the “equivalents” you can always
succeed in solving the problem desired.
We will now point out four transformations which are very easy to
effect, and result from the rule of “equivalents.”
1. Replacement of the two balls, situated on the same line and
separated by an empty cup, by one put into that cup. Thus I can replace
23 and 25 by a single ball at 24.
2. Suppression of tierces. And by the above movement I suppress the
tierce 9-16-23.
3. Correspondent “cases” are two holes situated in the same line and
separated by two cups. If two corresponding cups are filled, I can
suppress the balls which occupy them. So I can put aside 4 and 23.
[Illustration: Fig. 868.—Correspondents and equivalents.]
4. It is permissible to move a ball into one of the correspondent cups
if it be vacant; thus I can put 10 into 29.
These are the four transformations which can be made evident with the
rings, without displacing the balls. To do this we need have only seven
rings large enough to pass over the balls and to surround the holes in
which they rest. Let us take an example.
_Solitaire_ with 33 holes (fig. 869). _Final solution of the single
ball._
1st Vertical row: 7 and 21 are occupied, and the intermediate hole 14,
being empty, I place a ring upon 14.
2nd Vertical row: No. 8 takes 15, and comes into 22; I place a ring on
22.
3rd Vertical row: I suppress the corresponding balls, 4-23 and 16-31,
there now only remains 9, so I place a ring on 9.
4th Vertical row: I suppress the correspondents 10-29, put 2 into 17,
and I place a ring upon 17.
5th Vertical row: I suppress correspondents 6-25, put 33 into 18, and I
place a ring upon 18.
6th Vertical row: No. 12 takes 19 and comes to 26; I place a ring on 26.
7th Vertical row: No. 20 is the only ball; I place a ring on 20.
(It must be understood that these operations should be proceeded with
mentally; the balls must not be disturbed.)
We have thus reduced the problem to seven ringed balls, which are 14,
22, 9, 17, 18, 26, and 20 which are indicated on the diagram by the
line drawn through each vertically. They are all comprised in the three
horizontal rows, 3, 4, 5.
[Illustration: Fig. 869.—Single ball solution.]
We can now set to work upon these three rows in the same manner as
before, considering the rings as balls.
3rd row: We find (and leave) a ring upon 9.
4th row: The two corresponding rings, 17-20, neutralize each other, and
we suppress them. We carry 14 to 17, and take 17 with 18, which comes
into 16. We leave a ring on 16.
5th row: Carry the ring 26 to 23, take 23 with 22, which comes thus to
24, and we leave a ring on 24.
We now have reduced our problem to three rings, 9, 16, and 24, all in
the central square, indicated in the diagram by horizontal bars. It is
easy to see that 9 will take 16 and 24 and come into 25, and 25 will
remain alone—as was intended to be done—a single ball upon the board,
indicated by the circle around it in the cut.
By playing the “equivalent” method you will always arrive at this
result—a single ball in No. 25. It may now be perceived how we cannot
only arrive at a satisfactory solution, but by means of the rings
ascertain whether we shall succeed in our game without disturbing a
single ball. After some experience we may even learn to dispense with
the rings altogether.
[Illustration: Fig. 870.—Solitaire board.]
FOOTNOTES:
[41] From “La Nature.” Notice of M. Ch. Boutemps.