For this revised and enlarged edition, the author added material on such wide-ranging topics as the psychology of lottery players, the arrangement of chromosomes in a human cell, new and larger prime numbers, the fair division of a cake, how to find the shortest possible way to link a dozen locations by rail, and many other absorbing conundrums.
This appealing volume reflects the author’s longstanding concern with demonstrating the practical and concrete applications of mathematics as well as its theoretical aspects. It not only clearly and convincingly answers the question asked of Dr. Steinhaus but also offers readers a fascinating glimpse into the world of numbers and their uses.
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By H. Steinhaus
Dover Publications, Inc.Copyright © 1983 Morris Kline
All rights reserved.
Triangles, Squares, and Games
From these four small boards (1) we can compose a square or an equilateral triangle, according as we turn the handle up or down. The proof is given by sketch (2).
To decompose a square into two squares we draw a right triangle (3); to verify that the large square is the sum of the two others, we cut the medium square into four parts by a vertical and a horizontal line through its center, and shift these parts (without turning them) to cover the corners of the large square; the uncovered part of the large square is exactly the size of the small square. To verify this we have only to remark that a = b + c. The meaning of the theorem thus proved is clear when we look at the triangle (4) with sides 3, 4, and 5: 9 + 16 = 25. Thus we can draw a right angle by using a string 12 inches long with knots 3, 4, and 5 inches apart.
We may also verify this property of a right triangle without squares (5).
Let us draw equilateral triangles upon the sides of a given triangle ABC, one of whose angles (C) is equal to 60° (6). The combined area of the original ABC and of the new triangle opposite C is equal to the combined area of the remaining triangles. Proof(7): 1 + 2 + 3 = 1' + 2' + 3'.
To draw an equilateral triangle we can start with any triangle and trisect its angles: the little triangle in the middle is equilateral (8).
The trisection of an angle can be done very accurately by first halving it (9) and then dividing the chord of the half into three equal parts: the radius cutting 2/3 off the chord trisects the angle. This construction is only an approximate one.
It is easy to cover a plane with squares of different sizes (10). A very interesting problem is to divide a rectangle into squares, each of them different. On the following page they are given (11), nine in number, with sides 1, 4, 7, 8, 9, 10, 14, 15, 18. Problem: form a rectangle with them. This is the simplest example of division of a rectangle into different squares. A division into fewer than nine different squares is impossible.
It is possible to divide a square into different squares. One of the simplest cases is drawn here (12). The sides of the 24 squares are: 1, 2, 3, 4, 5, 8, 9, 14, 16, 18, 20, 29, 30, 31, 33, 35, 38, 39, 43, 51, 55, 56, 64, and 81. Can a square be decomposed into fewer than 24 different squares?
To cut out of any triangle another with an area equal to one-seventh of the whole, we divide (13) every side in the ratio 1:2 and connect the points of division with opposite vertices; the shaded area in the middle is one-seventh of the whole and the proof is to be read from the adjoining figure (14): the black and the shaded parts give 7 congruent triangles, each equal to the shaded area; as the 6 black triangles can be used to cover the white parts, the 7 congruent triangles together give the great triangle.
The simplest division of the plane, into equal squares (15), gives a board for many games. Two people can play 'three-in-a-row' on this (16) nine-square chessboard. One of the players has three white pieces, the other has three black ones. They place the pieces in turn, and when all six pieces are on the board, each may be moved to any adjoining square (but not diagonally). The one who first places his pieces in a horizontal, vertical, or diagonal row is the winner. The first player is sure to win if he at once occupies the center square and then plays sensibly. For, if White occupies e, Black can counter in only two ways: by covering either a corner square or a side square between corners. If Black covers a, White ought to cover h, compelling Black to choose b, then White will have to cover c, causing Black to occupy g. Now White in the next two moves will pass from e to f and from h to i and win. If Black begins by choosing b, White will cover g, Black c, White a, Black d, and White will pass from g to h and then from h to i, moves that the black piece covering c will be unable to prevent. If the leader is not allowed to cover e, the game, if played cleverly by both partners, will degenerate to an endless repetition of identical cycles.
There are positions in chess that permit of an exact analysis. For example, the end-game of Dr. J. Berger (17) assures victory to White, provided White begins with the move Q-QKt8. He will not win if he begins with any other move, provided Black defends himself sensibly. But if White begins with the above-mentioned move and continues properly, in eight moves the game should become an evident win for him. Certain end-games are famous because of their cleverly hidden solutions. Although
(18) is not of this class, it is by no means an easy task for the beginner to find out how White can checkmate in four moves at most.
Dr. K. Ebersz's end-game is of an entirely mathematical character (19). It can be proved rigorously that White will not allow Black's king to take any of his pawns, provided that he always moves to the square on which Black's king is then standing. He must therefore start by the move B-F. If he observes this rule, the game will end in a draw, but if he makes one false move, then Black can prevent him, if he chooses, from applying such tactics, and may break through X-Y or O-O. An interesting end-game would be one in which the moves of one player were exactly determined by those of his opponent, the game also ending in a draw, but the player who first departed from the rule would lose the game, provided his opponent played in a certain way that would also be fully determined.
There is no need for the reader to be a great chessplayer in order to secure in a simultaneous game against two chess champions A and B the result 1:1. It is only necessary that A play with white and B with black pieces, and A begin the game. The reader R repeats the first move of A on B's chessboard, thus starting the play against B. After B's countermove R transfers it as his answer against A on A's chessboard. Thus on both chessboards the same game will be played. On the first chessboard the result for R can only be 1, 0, or 1/2 and on the second chessboard 0, 1, or 1/2. Thus in each case R wins one point (1 + 0, 0 + 1 or 1/2 + 1/2), while A and B together win only one point as well.
Under the rules of chess, Black wins when he is able to call 'Checkmate,' meaning that White's king cannot avoid capture on the next move. The game is a draw if it has reached a situation making victory impossible for either player. There is also a situation called 'pat,' which makes necessary a suicide for one of the kings. Our sketch (20) shows a position that cannot be classified as a victory, or as a draw, or as a pat. The last piece to be moved was the Black knight. It is now White's turn; but it is impossible for White to move.
There is no mathematical theory of the game of chess, but there is one in certain simpler games. For example, in a box (21) there are 15 numbered tablets, and there is an empty space for one more. Lay the tablets in the box in any desired order (22) and then, by suitable moves, arrange them as they were originally ordered. The theory is as follows: let us call the vacant place '16'; then every arrangement of the tablets is a permutation of the numbers 1, 2, 3 ... 15, 16. Now, by writing these numbers first in their natural order 1 ... 16 and then appropriately interchanging them with their neighbors, every desired order can be obtained. For instance, to get the arrangement 2, 1, 3, 4, 5 ... 16, one interchange is needed. We call it a move. Some arrangements require an odd, some an even number of moves. If an arrangement is to be reached by an odd number of moves, it is impossible to get it by an even number of moves. Let us imagine the contrary: an arrangement produced by an even number of moves and the same arrangement produced by an odd number of moves. Starting with the natural arrangement, executing the even number of moves and then the odd number of moves but in the opposite direction, we should come back to the natural order. Thus in an odd number of moves we could pass from the natural order to itself. This is impossible because every move is an interchanging of two neighbors. Consider first only moves interchanging 5 with 6. The first move of this kind changes 56 into 65, the second one changes 65 into 56, and so on; as we must eventually re-establish the natural order 56, the number of the moves considered is even. The same reasoning applies to the pair 1-2, to the pair 2-3 ... and to the pair 15-16: for every pair there is an even number of moves which interchanges it. Thus the total number of moves employed to pass from the natural order back to itself is even, being a sum of even numbers. Thus we can classify all arrangements into two classes: the 'even' and the 'odd' arrangements. Let us consider the arrangements of tablets in the box as an arrangement of numbers, reading them down line by line. When we shift the tablets in the box, we can only interchange the vacant place '16' with one of its neighbors. If this neighbor is the right or the left one, the interchanging is a 'move' in the previous sense, as if all the horizontal lines formed one line. If, however, we interchange the tablet '16' with its upper or lower neighbor, the step is equivalent to interchanging two tablets that, in the total line, have the distance 4. Such an interchange requires 7 moves, i.e. 7 interchangings of neighbors. To solve our problem, we must in any case bring the tablet '16' in the box back to its initial position in the bottom right-hand corner; it must be therefore shifted as many times to the left as to the right and as many times up as down. The number of horizontal shiftings is therefore an even number 2h, and the number of vertical shiftings also an even number 2v. The whole process is thus equivalent to 2h moves plus 2v x 7 moves = 2h + 14v moves and this number is even. Consequently, if an arrangement is to be obtained from the basic one by an odd number of 'moves,' the problem of passing back is insoluble. For instance, we cannot, by moving the tablets, change the arrangement given on our illustration into that shown on the drawing of the box, nor can we pass from the first to the second one. (Why?) All arrangements that can be reached by an even number of 'moves' define soluble problems; the reader may try to prove this statement.
All the games mentioned here and many others have something in common. Not only the end-games of chess but also 'wolf and sheep' and 'three-in-a-row' have theories indicating which color (Black or White) will win, provided he plays properly. The theory teaches at the same time how to play properly. The case of a draw seems to be an exception but we can exclude it by the rule that the player who is confronted by a position that has previously occurred and who makes the same move a second time is defeated. Now, there is a general theorem to the effect that all games of the kind described above are either unfair or futile. We call a game futile if it permits a draw when properly played by both partners. In certain games no draw is possible; we call them categorical. In others we can exclude the draw by supplementary rules, as mentioned above. Our thesis affirms that all categorical games are unfair. The meaning of this thesis is that only one color has a method of winning whatever his opponent may do. To discover this method may be very easy, as in 'wolf and sheep,' or very difficult, as in some end-games of chess; nevertheless the existence of the winning color and of the winning method is certain. The theorem is general enough to apply also to such games as chess, provided it has been made categorical by the rule of repetition mentioned above and by considering as defeated a partner who gets into a 'rut.'
To prove the theorem, consider an end-game that assures a victory to White after 4 moves at most. Let us call it an EG4. It is clear that there exists an initial move for White so that whatever Black's answer may be, the resulting position becomes an EG3. Let us call this move a good one. White now has another good move, reducing the position to an EG2, and so on, until an EG, is reached. There is now a victorious move for White: the checkmate. Of course, a wrong defense by Black can accelerate his defeat; instead of being checkmated after exactly 4 moves he may be so after 3 moves. In any case White has a series of good moves leading to victory in 4 moves or less. Now it is clear what an EGn means. All EGn's, where n is any natural number (n = 1, 2, 3 ...), are called victorious for White. Let us consider the initial position in chess, when all 32 pieces are ordered ready for the battle. Two cases are logically possible and mutually exclusive: (I) the position is victorious for White, (II) the position is not victorious for White. In the first case, the game of chess is essentially victorious for White: it is simply an EGn. In the second case, the initial position is not an EGn. In this case there exists for every definite move M of White an answer such that the resulting position is not an EGn—in fact, if no such answer should exist, every answer would change the position into one victorious for White and, consequently, the initial position itself would be victorious for White, against our assumption. Thus we know that M can be answered by Black in a way that the resulting position is still not an EGn. Applying the same argument to the new position, we see that Black can find an answer to any second movement M' of White's, leading to a 'not an EGn.' As White can win only by getting an EG1—which never will happen—and as the game is categorical, Black can win it, whatever White may do. We don't know which of the two cases, I or II, corresponds to the real modified chess, but we are sure that one and only one of them is true, a fact implying that chess is an unfair game. The same reasoning applies to checkers, halma, and many other games. If they are not categorical, they must be futile. We do not know whether ordinary (not modified) chess is futile or not. In the negative case we know that it is unfair, but we do not know which color is the privileged one. Even if we knew it, we should not necessarily have the knowledge of the winning method. If we knew that chess is futile, we could still be ignorant of the methods leading to a draw.
There are games of a different kind to which the theory advanced above does not apply. The same board used for the 'three-in-a-row' game can be used for the following one. The board (23) carries 9 numbers, some black and some white. White writes 0, 00, or 000 on a piece of paper and on another Black writes I, II, or III, but neither of them sees what his opponent writes. Then they produce the scraps and determine the column and the line on the board. The number found in the line and column gives the number of dimes White is to get from his opponent, if it is white, and the number he has to pay to Black, if it is black. The peculiarity of this game consists in its not being 'closed.' To explain the meaning of this remark, let us suppose that White always chooses 00 and that Black has already noticed this preference. The best Black can do under such circumstances is to choose II; he will win three dimes by this method in every run. Of course White will learn this trick of Black by experience, and before long he will find that he has to change his habit and choose 0. This policy will bring him two dimes in every run, so long as Black keeps to II. It is easy to see that this mutual adaptation never leads to a rigid method for either of the partners. In chess the situation is different. In Dr. Berger's end-game, the solution given in our text is the best for both partners. If White knows that his opponent never makes mistakes, he will begin with Q—QKt8; otherwise he could not expect a victory in his 13th move. If Black knows his opponent is an ideal player, he will answer by B—QB5; any different move would enable White to checkmate him before the 13th move. The contest will thus continue according to the 'principal solution' of our test. In such a play the two methods, Black's and White's, are mutually best. The existence of such 'principal solutions' makes a game 'closed.' Chess is thus a closed game, and all ordinary games like checkers, halma, and so on which we have shown to be unfair or futile are closed, whereas our new game of 9 squares is neither closed nor unjust; it is open and equitable like 'matching pennies.' Of course, the fact that in these games there is no first and no second player is essential.
Excerpted from Mathematical Snapshots by H. Steinhaus. Copyright © 1983 Morris Kline. Excerpted by permission of Dover Publications, Inc..
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Table of Contents1. "Triangles, Squares, and Games"
2. "Rectangles, Numbers, and Tunes"
3. "Weighing, Measuring, and Fair Division"
4. "Tessellations, Mixing of Liquids, Measuring Areas and Lengths"
5. "Shortest Paths, Locating Schools, and Pursuing Ships"
6. "Straight Lines, Circles, Symmetry, and Optical Illusions"
7. "Cubes, Spiders, Honeycombs, and Bricks"
8. "Platonic Solids, Crystals, Bee's Heads, and Soap"
9. "Soap-Bubbles, Earth and Moon, Maps, and Dates"
10. "Squirrels, Screws, Candles, Tunes, and Shadows"
11. "Surfaces Made of Straight Lines, the Chain, the Toycart, and the Minimal Surface"
12. "Platonic Bodies Again, Crossing Bridges, Tying Knots, Coloring Maps, and Combing Hair"
13. "Board of Fortune, Frogs, Freshmen, and Sunflowers"