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 7s. 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 11s. 0d.—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.

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.

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.

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.

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).

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.

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—

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:—

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).

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.

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.

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.