Detailed solutions show several methods by which a particular problem may be answered, why one method is preferable, and where the others fail. With numerous worked examples, the clear, step-by-step analyses cover how the problem should be approached, including hints and enumeration of possibilities and determination of probabilities, application of the theory of probability, and evaluation of contingencies and mean values. Readers are certain to improve their puzzle-solving strategies as well as their mathematical skills.
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The Master Book of Mathematical Recreations
By Fred Schuh, T.H. O'Beirne, F. Göbel
Dover Publications, Inc.Copyright © 1968 Dover Publications, Inc.
All rights reserved.
HINTS FOR SOLVING PUZZLES
I. VARIOUS KINDS OF PUZZLES
1. Literary puzzles. There is a great difference between various puzzles, not only in their difficulty, but also in their essential nature. According to their character we can divide puzzles into two classes, which we shall call literary puzzles and pure puzzles.
To the literary puzzles belong crosswords, word-play riddles, proverbs that have to be guessed from certain data, and the like. Of course, skill in solving this kind of puzzle does depend on inborn shrewdness, but still more on knowledge of words and often also on geographical and historical knowledge. This knowledge can be supplemented to a large extent by using an atlas and, especially, an encyclopedia. However, if you do not have the necessary simple knowledge (of proverbs, of less common expressions, etc.), you will not succeed in cracking a somewhat difficult literary puzzle, because you will not be able to use the appropriate aids sufficiently well.
The peculiarity of a literary puzzle is that there is no sure way to attain the answer. You must have a flash of inspiration, or have the luck to guess a phrase from a few fragments, or something of the sort. Also, with puzzles of this kind (unless they are very easy) you can never know for certain that the solution found is the intended one, and whether it is the only solution, since trying out all possibilities is so time-consuming that, in many cases, this would take years (sometimes even centuries). However, usually you will have practical certainty that the intended solution has been found. For example, if part of the task is to find a proverb of some definite length, it would be extremely improbable that there was another proverb of precisely the same length which would also satisfy all given conditions, so improbable that the chance of this can be completely neglected (even though it cannot be numerically estimated).
There is a certain kind of literary puzzle where it is impossible to be sure of having obtained the best solution. As an example, suppose that it is required to compose a sentence of as many words as possible which all begin with the same letter. A second example is the problem of forming the longest possible closed chain of words with the property that the last syllable of each word is the same as the first syllable of the next word. Also, with a puzzle of this kind, the poser is under no obligation to have a more or less suitable solution available.
It is hardly or not at all possible to make general remarks (by way of advice) about the solution of literary puzzles, passing beyond what is done automatically by anyone who regularly solves such puzzles. So we shall not occupy ourselves with these puzzles any further.
2. Pure puzzles. While a literary puzzle is linked to a definite language, or sometimes to a few languages, this is not the case with a pure puzzle. With a pure puzzle, the question can be translated into any language, without the nature of the puzzle changing in any way. When solving a literary puzzle, you need some information that you cannot deduce on your own. With a pure puzzle, too, you often have to have a fund of knowledge, but this is mostly of such a nature that you yourself can discover what you need, if you are sufficiently intelligent.
A pure puzzle usually has a numerical, sometimes also a more or less geometrical content. The puzzle can be solved by pure reasoning alone. It is also typical of such a puzzle that it can always be decided with certainty whether the solution is correct. In many cases this can be determined easily and it is then practically out of the question that a wrong solution will be accepted in place of the correct one. Usually either you do not find a proper solution, or else you obtain the correct solution (or, it may be, one of the correct solutions), although it is always possible that persons lacking in self-criticism will consider a wrong solution to be correct.
Yet there is a kind of pure puzzle where an incorrect solution can be mistaken for a correct one, without this implying lack of self-criticism. As an example we mention the case of a puzzle which has more than one solution, where all solutions are required. It may also happen that the solution is required to be one for which a certain number (for example, the number of corners in a figure to be formed from dominoes that fit together in a certain manner) is as large or as small as possible. In such a case, the possibility exists that something has been overlooked, so that some solutions are found, but not all the solutions. The solution for which the number in question is maximum or minimum might happen to be among the solutions that have been overlooked; the supposed solution is then incorrect, without this necessarily implying a lack of intelligence.
Another example of the same nature occurs when the problem requires the smallest number of operations (for example, in moving cubes) by which a certain result (for example, a given final position of the cubes) can be obtained. We call these operations "moves." In a case like this it may happen that a certain move has been overlooked, or discarded too quickly because, considered superficially, it seemed unpromising.
3. Remarks on pure puzzles. No very sharp line can be drawn between pure puzzles and formal mathematics. In mathematics, too, questions occur that have more or less the character of a puzzle. Whether one speaks of a mathematical problem or of a puzzle, in such a case, depends on the significance of the question (and, to some extent, also on the nature of the reasoning that leads to the solution); and it is partially a matter of taste as well. On the other hand, there are problems that impress everyone as puzzles, but which still allow a fruitful application of mathematics (in particular, of arithmetic). The greater your skill in arithmetic, the less time it will take you to find the solution. The puzzles with squares in §§229 and 230 provide a striking example of this.
A puzzle can be considered as uninteresting when the data and the question can without difficulty be put into the form of equations by anyone who has some knowledge of algebra, to let the unknown (s) be found from solution of the equation (s). A mathematician is not in the least interested in such a puzzle, whereas a good puzzle should satisfy the requirement that it can arouse an experienced mathematician's interest, too. As an example of a puzzle which ought rather to be called a dull problem in algebra, let John ask Peter to think of a number, add 15 to it, multiply the sum by 3, subtract 6 from the product, and divide the difference by 3. Peter then has to announce the quotient, after which John is to determine what number Peter had thought of. Obviously it is 13 less than the quotient mentioned by Peter. The puzzle may be modified by requiring Peter to subtract his first number from the quotient obtained. Without Peter needing to say anything, John will then be able to say that the final result is 13; however, now it is impossible for John to know what number Peter had thought of, since every number leads to 13 after the operations in question.
How little interest these puzzles have, is also evident from the fact that they can be varied endlessly. One can make them as complicated as one pleases, and also allow the other person to think of more than one number.
4. Puzzle games. We imagine a game that is played by two persons, John and Peter; we assume that luck plays no part (no more than with chess or draughts). John and Peter make moves in turn, and the winner is the one who succeeds in achieving a certain result. Often such a game is so complicated (again I mention chess and draughts) that it is impossible for a human being to analyze it completely. The game is then different from a puzzle; making a complete analysis could be called a puzzle, but as this puzzle is unsolvable (in the sense that it is too difficult), it has to be left out of consideration.
If the game is so simple that it can have a complete analysis, we speak of a puzzle game. As an example we mention the well-known children's game of noughts and crosses, which will be discussed in Chapter III. This will show that surprisingly many elegant combinations are contained even in such a seemingly very simple game. The laboriousness of the complete treatment of this game shows quite clearly the impossibility of similarly disposing of much more complicated games like chess and draughts. Chapter III can further be considered as an example of a complete analysis of a puzzle game.
The interest of a puzzle game is, of course, lost when both players have seen through it completely. However, a game like chess will never lose interest in this way, although several puzzles and puzzle games can be derived from it. An endgame that is not too complicated can be considered as a puzzle game. A related puzzle is the problem of the smallest number of moves in which mate (or sometimes the promotion of a pawn) can be enforced by White; here it is assumed, of course, that Black puts up the strongest possible defense, playing in such a way that the number of moves becomes as large as possible.
The last-named puzzle offers another example of a case in which one can easily mistake an incorrect answer for a correct one (cf. §2). For, in a not too simple endgame, it may very well happen that even an experienced chessplayer will wrongly consider a certain sequence of moves to be the shortest possible one.
5. Correspondences and differences between puzzles and games. A game like chess shows a resemblance to puzzle solving in many situations, in particular (as we said before) in the endgame, but also when, through a combination, a certain result has to be achieved that is no way near to a final result. A chess problem (mate in so many moves) is nothing else but a puzzle.
In the opening and frequently also in the middle game, chess cannot be compared to puzzles. There the evaluation of the position plays an important role, at a stage when it is not possible to appreciate exactly how an advantageous position can be turned into victory. Besides, in chess quite different abilities of the players manifest themselves, and, as a consequence, a good chess player is not necessarily an able solver or composer of chess problems, and conversely. Naturally, these two abilities are not completely foreign to each other. A good chess player needs a great talent for combination — supplemented by a good ability to visualize situations (the latter especially when playing blindfold chess). He also needs a good memory to be able to retain the combinations — as well as a feeling for position, a quick activity of the mind to escape from pressures of time, and a strong power of concentration, to enable him to put aside all other things at any moment, with the ability not to have his attention diverted by anything or anybody. Added to this, there is the psychological factor, which consists of knowing and understanding the opponent's ways of playing and his weaknesses, so that a good chess player will play differently against one player and against another, in the same situation. As a consequence of all this, chess playing and puzzle solving can hardly be put on one level. In chess, you have a flesh-and-blood opponent, whose strong and weak points you are well-advised to know. In solving puzzles, if you have an opponent at all, he is imaginary; hence here you are your own opponent, in the end.
The requirements for chess and puzzle solving diverge to such an extent that the puzzle solver in a person may hamper the chess player, and conversely. A good puzzle solver, who has acquired the habit of scrupulously examining all possibilities, runs the risk of wanting to do the same thing in practical chess, where he gets entangled in the multitude of possibilities, is unable to make a choice, and then finally, being pressed for time, makes a bad move. Conversely, an experienced chess player who wishes to solve a puzzle runs the risk of relying too much on his intuition, leaving many possibilities unexamined; sometimes he will do this successfully and find the solution of the puzzle quickly, but often he will not.
With regard to games, in what follows we shall occupy ourselves only with those to which the name "puzzle game" applies.
II. SOLVING BY TRIAL
6. Trial and error. Trial is an activity that is important and useful when solving a pure puzzle. This may invite the belief that solving a pure puzzle is a matter of accident, hence of luck; all that matters, it might be argued, is whether one tries the correct thing earlier or later. This opinion is very widespread, but quite incorrect, at least with regard to good puzzles, by which we mean puzzles which cannot in practice be exhausted by trial, to the extent of finding all the solutions. In solving a puzzle in which an excessive role is played by the element of trial, notwithstanding all ingenuity, you may be lucky; but if you are lucky enough to find a solution, you still do not know whether this is the only solution, let alone how many solutions there are. If these questions form no part of the problem, and you are only required to find some solution, then luck can indeed play its part.
In such a case, however, one should speak of a bad puzzle. Such a puzzle can be composed (that is, put to others) by someone who has no idea of puzzles himself. We should like to illustrate this with an example. Consider a figure composed of a number — 40, say — of adjacent regular hexagons. Each of these hexagons is divided into 6 equilateral triangles by 3 diagonals connecting opposite vertices. Two triangles which have a side in common and which belong to different hexagons (which occurs a large number of times in the figure) are painted with the same color. Altogether 3 or 4 colors, say, have been used, while the distribution of colors among the pairs of triangles is quite irregular. Now one cuts out the hexagons and interchanges them in a random way, also giving arbitrary rotations to individual hexagons. The figure thus obtained (with the hexagons in new positions, connected to each other again) is now presented as a puzzle. The problem would then be to cut out the hexagons and rearrange them to produce a figure with the same boundary, with like colors making contact everywhere. The person who poses the puzzle knows that success is possible, but knows nothing of other solutions, when there may be many, perhaps millions. Such a thing is wrong. There is no proper idea behind the puzzle.
The opinion that solving a puzzle is a matter of luck is usually found among people who have only a superficial interest in puzzles, and who have rarely or never taken the trouble to solve a good puzzle by reasoning. The same opinion is sometimes held by people who do take pleasure in solving puzzles, but who are not sufficiently intelligent to replace — or, at any rate, supplement — haphazard trial and error by reasoning. Such a puzzle solver often does not remember which cases have already been tried, and which have not; he does the same job several times and overlooks other possibilities altogether. It is clear that this form of puzzle solving will not guarantee success.
Admittedly, of course, luck may play a role even in solving a pure puzzle by reasoning, if one happens to hit upon an efficient procedure. However, the same thing may occur in purely scientific work. Here, in place of speaking of luck, it would be better to speak of an intuition for finding the right path, or a feeling for the right method.
7. Systematic trial. Solving a pure puzzle by trial has to be done with a fair amount of discretion. One might call it reasoned trial or judicious trial, but the intention is perhaps best represented by the expression "systematic trial." This is trial made in such a way that at any moment you know precisely which cases have already been considered, and which have not, and so at a given moment you are able to say, "Now I have examined all possibilities; these are the solutions and there can be no others."
Of course, when the puzzle is somewhat complicated, and you have had to consider many possibilities, it is not always out of the question that you have overlooked some possibility, but such a thing can also occur in solving a mathematical problem. In these cases, the error made is not due to the method, but to incidental carelessness. Without systematic trial, one makes errors systematically, in a sense, and often one goes round in a circle without noticing it.
In a systematic trial you should record accurately which possibilities have been examined, and also which solutions have arisen from one possibility, and which from another. Only in this way will you be able to check the given solution afterwards and, if necessary, improve upon it, with regard either to the correctness of the results or to the simplicity of the method employed.
8. Division into cases. We would now like to explain how to organize a systematic trial. You begin by dividing the various possibilities into groups; in other words, you begin by successively making various assumptions, each of which represents a group of possibilities. Here it is important to arrange these assumptions according to some system, to ensure that no assumption and no group of possibilities is left unexamined. As an example we take a digit puzzle in which there are certain places where unknown digits have to be filled in; these digits have to satisfy certain conditions which are mentioned in the puzzle. Here you consider some fixed place, and successively work out the ten assumptions that a digit 0, 1,2, 3, 4, 5, 6, 7, 8, or 9 occurs in that position.
Excerpted from The Master Book of Mathematical Recreations by Fred Schuh, T.H. O'Beirne, F. Göbel. Copyright © 1968 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
ContentsChapter I: Hints for Solving Puzzles,
Chapter II: Some Domino Puzzles,
Chapter III: The Game of Noughts and Crosses,
Chapter IV: Number Systems,
Chapter V: Some Puzzles Related to Number Systems,
Chapter VI: Games with Piles of Matches,
Chapter VII: Enumeration of Possibilities and the Determination of Probabilities,
Chapter VIII: Some Applications of the Theory of Probability,
Chapter IX: Evaluation of Contingencies and Mean Values,
Chapter X: Some Games of Encirclement,
Chapter XI: Sliding-Movement Puzzles,
Chapter XII: Subtraction Games,
Chapter XIII: Puzzles with Some Mathematical Aspects,
Chapter XIV: Puzzles of Assorted Types,
Chapter XV: Puzzles in Mechanics,