Amusements in Mathematics Part 11

285.--THE FOUR POSTAGE STAMPS.

+---+----+----+----+ | 1 | 2 | 3 | 4 | +---+----+----+----+ | 5 | 6 | 7 | 8 | +---+----+----+----+ | 9 | 10 | 11 | 12 | +---+----+----+----+ "It is as easy as counting," is an expression one sometimes hears. But mere counting may be puzzling at times. Take the following simple example. Suppose you have just bought twelve postage stamps, in this form--three by four--and a friend asks you to oblige him with four stamps, all joined together--no stamp hanging on by a mere corner. In how many different ways is it possible for you to tear off those four stamps? You see, you can give him 1, 2, 3, 4, or 2, 3, 6, 7, or 1, 2, 3, 6, or 1, 2, 3, 7, or 2, 3, 4, 8, and so on. Can you count the number of different ways in which those four stamps might be delivered? There are not many more than fifty ways, so it is not a big count. Can you get the exact number?

286.--PAINTING THE DIE.

In how many different ways may the numbers on a single die be marked, with the only condition that the 1 and 6, the 2 and 5, and the 3 and 4 must be on opposite sides? It is a simple enough question, and yet it will puzzle a good many people.

287.--AN ACROSTIC PUZZLE.

In the making or solving of double acrostics, has it ever occurred to you to consider the variety and limitation of the pair of initial and final letters available for cross words? You may have to find a word beginning with A and ending with B, or A and C, or A and D, and so on. Some combinations are obviously impossible--such, for example, as those with Q at the end. But let us a.s.sume that a good English word can be found for every case. Then how many possible pairs of letters are available?

CHESSBOARD PROBLEMS.

"You and I will goe to the chesse."

GREENE'S _Groatsworth of Wit._ During a heavy gale a chimney-pot was hurled through the air, and crashed upon the pavement just in front of a pedestrian. He quite calmly said, "I have no use for it: I do not smoke." Some readers, when they happen to see a puzzle represented on a chessboard with chess pieces, are apt to make the equally inconsequent remark, "I have no use for it: I do not play chess." This is largely a result of the common, but erroneous, notion that the ordinary chess puzzle with which we are familiar in the press (dignified, for some reason, with the name "problem") has a vital connection with the game of chess itself. But there is no condition in the game that you shall checkmate your opponent in two moves, in three moves, or in four moves, while the majority of the positions given in these puzzles are such that one player would have so great a superiority in pieces that the other would have resigned before the situations were reached. And the solving of them helps you but little, and that quite indirectly, in playing the game, it being well known that, as a rule, the best "chess problemists" are indifferent players, and vice versa. Occasionally a man will be found strong on both subjects, but he is the exception to the rule.

Yet the simple chequered board and the characteristic moves of the pieces lend themselves in a very remarkable manner to the devising of the most entertaining puzzles. There is room for such infinite variety that the true puzzle lover cannot afford to neglect them. It was with a view to securing the interest of readers who are frightened off by the mere presentation of a chessboard that so many puzzles of this cla.s.s were originally published by me in various fanciful dresses. Some of these posers I still retain in their disguised form; others I have translated into terms of the chessboard. In the majority of cases the reader will not need any knowledge whatever of chess, but I have thought it best to a.s.sume throughout that he is acquainted with the terminology, the moves, and the notation of the game.

I first deal with a few questions affecting the chessboard itself; then with certain statical puzzles relating to the Rook, the Bishop, the Queen, and the Knight in turn; then dynamical puzzles with the pieces in the same order; and, finally, with some miscellaneous puzzles on the chessboard. It is hoped that the formulae and tables given at the end of the statical puzzles will be of interest, as they are, for the most part, published for the first time.

THE CHESSBOARD.

"Good company's a chessboard." BYRON'S Don Juan, xiii. 89.

A chessboard is essentially a square plane divided into sixty-four smaller squares by straight lines at right angles. Originally it was not chequered (that is, made with its rows and columns alternately black and white, or of any other two colours), and this improvement was introduced merely to help the eye in actual play. The utility of the chequers is unquestionable. For example, it facilitates the operation of the bishops, enabling us to see at the merest glance that our king or p.a.w.ns on black squares are not open to attack from an opponent's bishop running on the white diagonals. Yet the chequering of the board is not essential to the game of chess. Also, when we are propounding puzzles on the chessboard, it is often well to remember that additional interest may result from "generalizing" for boards containing any number of squares, or from limiting ourselves to some particular chequered arrangement, not necessarily a square. We will give a few puzzles dealing with chequered boards in this general way.

288.--CHEQUERED BOARD DIVISIONS.

I recently asked myself the question: In how many different ways may a chessboard be divided into two parts of the same size and shape by cuts along the lines dividing the squares? The problem soon proved to be both fascinating and bristling with difficulties. I present it in a simplified form, taking a board of smaller dimensions.

[Ill.u.s.tration: +---+---*---+---+ +---+---+---*---+ +---+---+---*---+ | | H | | | | | H | | | | H | +---+---*---+---+ +---+---*===*---+ +---*===*---*---+ | | H | | | | H | | | H H H | +---+---*---+---+ +---+---*---+---+ +---*---*---*---+ | | H | | | | H | | | H H H | +---+---*---+---+ +---*===*---+---+ +---*---*===*---+ | | H | | | H | | | | H | | | +---+---*---+---+ +---*---+---+---+ +---*---+---+---+ +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ | | | | | | | +---+---+---+---+---+---+ +---+---*---+---+ +---+---+---*---+ +---+---+---*---+ | | H | | | | | H | | | | H | +---*===*---+---+ +---*===*===*---+ +---+---*===*---+ | H | | | | H | | | | | H | | +---*===*===*---+ +---*===*===*---+ +---+---*---+---+ | | | H | | | | H | | | H | | +---+---*===*---+ +---*===*===*---+ +---*===*---+---+ | | H | | | H | | | | H | | | +---+---*---+---+ +---*---+---+---+ +---*---+---+---+ ]

It is obvious that a board of four squares can only be so divided in one way--by a straight cut down the centre--because we shall not count reversals and reflections as different. In the case of a board of sixteen squares--four by four--there are just six different ways. I have given all these in the diagram, and the reader will not find any others. Now, take the larger board of thirty-six squares, and try to discover in how many ways it may be cut into two parts of the same size and shape.

289.--LIONS AND CROWNS.

The young lady in the ill.u.s.tration is confronted with a little cutting-out difficulty in which the reader may be glad to a.s.sist her. She wishes, for some reason that she has not communicated to me, to cut that square piece of valuable material into four parts, all of exactly the same size and shape, but it is important that every piece shall contain a lion and a crown. As she insists that the cuts can only be made along the lines dividing the squares, she is considerably perplexed to find out how it is to be done. Can you show her the way? There is only one possible method of cutting the stuff.

[Ill.u.s.tration: +-+-+-+-+-+-+ | | | | | | | +-+-+-+-+-+-+ | |L|L|L| | | +-+-+-+-+-+-+ | | |C|C| | | +-+-+-+-+-+-+ | | |C|C| | | +-+-+-+-+-+-+ |L| | | | | | +-+-+-+-+-+-+ | | | | | | | +-+-+-+-+-+-+ ]

290.--BOARDS WITH AN ODD NUMBER OF SQUARES.

We will here consider the question of those boards that contain an odd number of squares. We will suppose that the central square is first cut out, so as to leave an even number of squares for division. Now, it is obvious that a square three by three can only be divided in one way, as shown in Fig. 1. It will be seen that the pieces A and B are of the same size and shape, and that any other way of cutting would only produce the same shaped pieces, so remember that these variations are not counted as different ways. The puzzle I propose is to cut the board five by five (Fig. 2) into two pieces of the same size and shape in as many different ways as possible. I have shown in the ill.u.s.tration one way of doing it. How many different ways are there altogether? A piece which when turned over resembles another piece is not considered to be of a different shape.

[Ill.u.s.tration: +---*---+---+ | H | | +---*===*---+ | HHHHH | +---*===*---+ | | H | +---+---*---+ Fig 1]

[Ill.u.s.tration: +---+---+---+---+---+ | | | | | | *===*===*===*---+---+ | | | H | | +---+---*===*---+---+ | | HHHHH | | +---+---*===*---+---+ | | H | | | +---+---*===*===*===* | H | | | | +---*---+---+---+---+ Fig 2]

291.--THE GRAND LAMA'S PROBLEM.

Once upon a time there was a Grand Lama who had a chessboard made of pure gold, magnificently engraved, and, of course, of great value. Every year a tournament was held at Lha.s.sa among the priests, and whenever any one beat the Grand Lama it was considered a great honour, and his name was inscribed on the back of the board, and a costly jewel set in the particular square on which the checkmate had been given. After this sovereign pontiff had been defeated on four occasions he died--possibly of chagrin.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | * | | | | | | | | +---+---+---+---+---+---+---+---+ | | * | | | | | | | +---+---+---+---+---+---+---+---+ | | | * | | | | | | +---+---+---+---+---+---+---+---+ | | | | * | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ ]

Now the new Grand Lama was an inferior chess-player, and preferred other forms of innocent amus.e.m.e.nt, such as cutting off people's heads. So he discouraged chess as a degrading game, that did not improve either the mind or the morals, and abolished the tournament summarily. Then he sent for the four priests who had had the effrontery to play better than a Grand Lama, and addressed them as follows: "Miserable and heathenish men, calling yourselves priests! Know ye not that to lay claim to a capacity to do anything better than my predecessor is a capital offence? Take that chessboard and, before day dawns upon the torture chamber, cut it into four equal parts of the same shape, each containing sixteen perfect squares, with one of the gems in each part! If in this you fail, then shall other sports be devised for your special delectation. Go!" The four priests succeeded in their apparently hopeless task. Can you show how the board may be divided into four equal parts, each of exactly the same shape, by cuts along the lines dividing the squares, each part to contain one of the gems?

292.--THE ABBOT'S WINDOW.

[Ill.u.s.tration]

Once upon a time the Lord Abbot of St. Edmondsbury, in consequence of "devotions too strong for his head," fell sick and was unable to leave his bed. As he lay awake, tossing his head restlessly from side to side, the attentive monks noticed that something was disturbing his mind; but n.o.body dared ask what it might be, for the abbot was of a stern disposition, and never would brook inquisitiveness. Suddenly he called for Father John, and that venerable monk was soon at the bedside.

"Father John," said the Abbot, "dost thou know that I came into this wicked world on a Christmas Even?"

The monk nodded a.s.sent.

"And have I not often told thee that, having been born on Christmas Even, I have no love for the things that are odd? Look there!"

The Abbot pointed to the large dormitory window, of which I give a sketch. The monk looked, and was perplexed.

"Dost thou not see that the sixty-four lights add up an even number vertically and horizontally, but that all the diagonal lines, except fourteen are of a number that is odd? Why is this?"

"Of a truth, my Lord Abbot, it is of the very nature of things, and cannot be changed."

"Nay, but it shall be changed. I command thee that certain of the lights be closed this day, so that every line shall have an even number of lights. See thou that this be done without delay, lest the cellars be locked up for a month and other grievous troubles befall thee."

Father John was at his wits' end, but after consultation with one who was learned in strange mysteries, a way was found to satisfy the whim of the Lord Abbot. Which lights were blocked up, so that those which remained added up an even number in every line horizontally, vertically, and diagonally, while the least possible obstruction of light was caused?

293.--THE CHINESE CHESSBOARD.

Into how large a number of different pieces may the chessboard be cut (by cuts along the lines only), no two pieces being exactly alike? Remember that the arrangement of black and white const.i.tutes a difference. Thus, a single black square will be different from a single white square, a row of three containing two white squares will differ from a row of three containing two black, and so on. If two pieces cannot be placed on the table so as to be exactly alike, they count as different. And as the back of the board is plain, the pieces cannot be turned over.

294.--THE CHESSBOARD SENTENCE.

[Ill.u.s.tration]

I once set myself the amusing task of so dissecting an ordinary chessboard into letters of the alphabet that they would form a complete sentence. It will be seen from the ill.u.s.tration that the pieces a.s.sembled give the sentence, "CUT THY LIFE," with the stops between. The ideal sentence would, of course, have only one full stop, but that I did not succeed in obtaining.

The sentence is an appeal to the transgressor to cut himself adrift from the evil life he is living. Can you fit these pieces together to form a perfect chessboard?

STATICAL CHESS PUZZLES.

"They also serve who only stand and wait." MILTON.

295.--THE EIGHT ROOKS.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | R | R | R | R | R | R | R | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ FIG. 1.]

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | R | | | | | | | | +---+---+---+---+---+---+---+---+ | | R | | | | | | | +---+---+---+---+---+---+---+---+ | | | R | | | | | | +---+---+---+---+---+---+---+---+ | | | | R | | | | | +---+---+---+---+---+---+---+---+ | | | | | R | | | | +---+---+---+---+---+---+---+---+ | | | | | | R | | | +---+---+---+---+---+---+---+---+ | | | | | | | R | | +---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+ FIG. 2.]

It will be seen in the first diagram that every square on the board is either occupied or attacked by a rook, and that every rook is "guarded" (if they were alternately black and white rooks we should say "attacked") by another rook. Placing the eight rooks on any row or file obviously will have the same effect. In diagram 2 every square is again either occupied or attacked, but in this case every rook is unguarded. Now, in how many different ways can you so place the eight rooks on the board that every square shall be occupied or attacked and no rook ever guarded by another? I do not wish to go into the question of reversals and reflections on this occasion, so that placing the rooks on the other diagonal will count as different, and similarly with other repet.i.tions obtained by turning the board round.

296.--THE FOUR LIONS.

The puzzle is to find in how many different ways the four lions may be placed so that there shall never be more than one lion in any row or column. Mere reversals and reflections will not count as different. Thus, regarding the example given, if we place the lions in the other diagonal, it will be considered the same arrangement. For if you hold the second arrangement in front of a mirror or give it a quarter turn, you merely get the first arrangement. It is a simple little puzzle, but requires a certain amount of careful consideration.

[Ill.u.s.tration +---+---+---+---+ | L | | | | +---+---+---+---+ | | L | | | +---+---+---+---+ | | | L | | +---+---+---+---+ | | | | L | +---+---+---+---+ ]

297.--BISHOPS--UNGUARDED.

Place as few bishops as possible on an ordinary chessboard so that every square of the board shall be either occupied or attacked. It will be seen that the rook has more scope than the bishop: for wherever you place the former, it will always attack fourteen other squares; whereas the latter will attack seven, nine, eleven, or thirteen squares, according to the position of the diagonal on which it is placed. And it is well here to state that when we speak of "diagonals" in connection with the chessboard, we do not limit ourselves to the two long diagonals from corner to corner, but include all the shorter lines that are parallel to these. To prevent misunderstanding on future occasions, it will be well for the reader to note carefully this fact.

298.--BISHOPS--GUARDED.

Now, how many bishops are necessary in order that every square shall be either occupied or attacked, and every bishop guarded by another bishop? And how may they be placed?

299.--BISHOPS IN CONVOCATION.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | B | B | B | B | B | B | B | B | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | B | B | B | B | B | B | | +---+---+---+---+---+---+---+---+ ]

The greatest number of bishops that can be placed at the same time on the chessboard, without any bishop attacking another, is fourteen. I show, in diagram, the simplest way of doing this. In fact, on a square chequered board of any number of squares the greatest number of bishops that can be placed without attack is always two less than twice the number of squares on the side. It is an interesting puzzle to discover in just how many different ways the fourteen bishops may be so placed without mutual attack. I shall give an exceedingly simple rule for determining the number of ways for a square chequered board of any number of squares.

300.--THE EIGHT QUEENS.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | | | | ..Q | | | | +---+---+---+...+---+---+---+---+ | | ..Q.. | | | | | +---+...+---+---+---+---+---+---+ | Q.. | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | Q | | +---+---+---+---+---+---+---+---+ | | Q | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | ..Q | +---+---+---+---+---+---+...+---+ | | | | | ..Q.. | | +---+---+---+---+...+---+---+---+ | | | | Q.. | | | | +---+---+---+---+---+---+---+---+ ]

The queen is by far the strongest piece on the chessboard. If you place her on one of the four squares in the centre of the board, she attacks no fewer than twenty-seven other squares; and if you try to hide her in a corner, she still attacks twenty-one squares. Eight queens may be placed on the board so that no queen attacks another, and it is an old puzzle (first proposed by Nauck in 1850, and it has quite a little literature of its own) to discover in just how many different ways this may be done. I show one way in the diagram, and there are in all twelve of these fundamentally different ways. These twelve produce ninety-two ways if we regard reversals and reflections as different. The diagram is in a way a symmetrical arrangement. If you turn the page upside down, it will reproduce itself exactly; but if you look at it with one of the other sides at the bottom, you get another way that is not identical. Then if you reflect these two ways in a mirror you get two more ways. Now, all the other eleven solutions are non-symmetrical, and therefore each of them may be presented in eight ways by these reversals and reflections. It will thus be seen why the twelve fundamentally different solutions produce only ninety-two arrangements, as I have said, and not ninety-six, as would happen if all twelve were non-symmetrical. It is well to have a clear understanding on the matter of reversals and reflections when dealing with puzzles on the chessboard.

Can the reader place the eight queens on the board so that no queen shall attack another and so that no three queens shall be in a straight line in any oblique direction? Another glance at the diagram will show that this arrangement will not answer the conditions, for in the two directions indicated by the dotted lines there are three queens in a straight line. There is only one of the twelve fundamental ways that will solve the puzzle. Can you find it?

301.--THE EIGHT STARS.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+ | |///| | | | |///| * | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | | |///|///| | | | +---+---+---+---+---+---+---+---+ | | |///| | |///| | | +---+---+---+---+---+---+---+---+ | |///| | | | |///| | +---+---+---+---+---+---+---+---+ |///| | | | | | |///| +---+---+---+---+---+---+---+---+ ]

The puzzle in this case is to place eight stars in the diagram so that no star shall be in line with another star horizontally, vertically, or diagonally. One star is already placed, and that must not be moved, so there are only seven for the reader now to place. But you must not place a star on any one of the shaded squares. There is only one way of solving this little puzzle.

302.--A PROBLEM IN MOSAICS.

The art of producing pictures or designs by means of joining together pieces of hard substances, either naturally or artificially coloured, is of very great antiquity. It was certainly known in the time of the Pharaohs, and we find a reference in the Book of Esther to "a pavement of red, and blue, and white, and black marble." Some of this ancient work that has come down to us, especially some of the Roman mosaics, would seem to show clearly, even where design is not at first evident, that much thought was bestowed upon apparently disorderly arrangements. Where, for example, the work has been produced with a very limited number of colours, there are evidences of great ingenuity in preventing the same tints coming in close proximity. Lady readers who are familiar with the construction of patchwork quilts will know how desirable it is sometimes, when they are limited in the choice of material, to prevent pieces of the same stuff coming too near together. Now, this puzzle will apply equally to patchwork quilts or tesselated pavements.

It will be seen from the diagram how a square piece of flooring may be paved with sixty-two square tiles of the eight colours violet, red, yellow, green, orange, purple, white, and blue (indicated by the initial letters), so that no tile is in line with a similarly coloured tile, vertically, horizontally, or diagonally. Sixty-four such tiles could not possibly be placed under these conditions, but the two shaded squares happen to be occupied by iron ventilators.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | V | R | Y | G | O | P | W | B | +---+---+---+---+---+---+---+---+ | W | B | O | P | Y | G | V | R | +---+---*===*---+---*===*---+---+ | G | P H W H V | B H R H Y | O | +---+---*===*---+---*===*---+---+ | R | Y | B | O | G | V | P | W | +---+---+---+---+---+---+---+---+ | B | G | R | Y | P | W | O | V | +---+---+---+---+---+---+---+---+ | O | V | P | W | R | Y | B | G | +---+---+---+---+---+---+---+---+ | P | W | G | B | V | O | R | Y | +---+---+---+---+---+---+---+---+ |///| O | V | R | W | B | G |///| +---+---+---+---+---+---+---+---+ ]

The puzzle is this. These two ventilators have to be removed to the positions indicated by the darkly bordered tiles, and two tiles placed in those bottom corner squares. Can you readjust the thirty-two tiles so that no two of the same colour shall still be in line?

303.--UNDER THE VEIL.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | | | V | E | I | L | | | +---+---+---+---+---+---+---+---+ | | | I | L | V | E | | | +---+---+---+---+---+---+---+---+ | I | V | | | | | L | E | +---+---+---+---+---+---+---+---+ | L | E | | | | | I | V | +---+---+---+---+---+---+---+---+ | V | I | | | | | E | L | +---+---+---+---+---+---+---+---+ | E | L | | | | | V | I | +---+---+---+---+---+---+---+---+ | | | E | V | L | I | | | +---+---+---+---+---+---+---+---+ | | | L | I | E | V | | | +---+---+---+---+---+---+---+---+ ]

If the reader will examine the above diagram, he will see that I have so placed eight V's, eight E's, eight I's, and eight L's in the diagram that no letter is in line with a similar one horizontally, vertically, or diagonally. Thus, no V is in line with another V, no E with another E, and so on. There are a great many different ways of arranging the letters under this condition. The puzzle is to find an arrangement that produces the greatest possible number of four-letter words, reading upwards and downwards, backwards and forwards, or diagonally. All repet.i.tions count as different words, and the five variations that may be used are: VEIL, VILE, LEVI, LIVE, and EVIL.

This will be made perfectly clear when I say that the above arrangement scores eight, because the top and bottom row both give VEIL; the second and seventh columns both give VEIL; and the two diagonals, starting from the L in the 5th row and E in the 8th row, both give LIVE and EVIL. There are therefore eight different readings of the words in all.

This difficult word puzzle is given as an example of the use of chessboard a.n.a.lysis in solving such things. Only a person who is familiar with the "Eight Queens" problem could hope to solve it.

304.--BACHET'S SQUARE.

One of the oldest card puzzles is by Claude Caspar Bachet de Meziriac, first published, I believe, in the 1624 edition of his work. Rearrange the sixteen court cards (including the aces) in a square so that in no row of four cards, horizontal, vertical, or diagonal, shall be found two cards of the same suit or the same value. This in itself is easy enough, but a point of the puzzle is to find in how many different ways this may be done. The eminent French mathematician A. Labosne, in his modern edition of Bachet, gives the answer incorrectly. And yet the puzzle is really quite easy. Any arrangement produces seven more by turning the square round and reflecting it in a mirror. These are counted as different by Bachet.

Note "row of four cards," so that the only diagonals we have here to consider are the two long ones.

305.--THE THIRTY-SIX LETTER-BLOCKS.

[Ill.u.s.tration]

The ill.u.s.tration represents a box containing thirty-six letter-blocks. The puzzle is to rearrange these blocks so that no A shall be in a line vertically, horizontally, or diagonally with another A, no B with another B, no C with another C, and so on. You will find it impossible to get all the letters into the box under these conditions, but the point is to place as many as possible. Of course no letters other than those shown may be used.

306.--THE CROWDED CHESSBOARD.

[Ill.u.s.tration]

The puzzle is to rearrange the fifty-one pieces on the chessboard so that no queen shall attack another queen, no rook attack another rook, no bishop attack another bishop, and no knight attack another knight. No notice is to be taken of the intervention of pieces of another type from that under consideration--that is, two queens will be considered to attack one another although there may be, say, a rook, a bishop, and a knight between them. And so with the rooks and bishops. It is not difficult to dispose of each type of piece separately; the difficulty comes in when you have to find room for all the arrangements on the board simultaneously.

307.--THE COLOURED COUNTERS.

[Ill.u.s.tration]

The diagram represents twenty-five coloured counters, Red, Blue, Yellow, Orange, and Green (indicated by their initials), and there are five of each colour, numbered 1, 2, 3, 4, and 5. The problem is so to place them in a square that neither colour nor number shall be found repeated in any one of the five rows, five columns, and two diagonals. Can you so rearrange them?

308.--THE GENTLE ART OF STAMP-LICKING.

The Insurance Act is a most prolific source of entertaining puzzles, particularly entertaining if you happen to be among the exempt. One's initiation into the gentle art of stamp-licking suggests the following little poser: If you have a card divided into sixteen s.p.a.ces (4 4), and are provided with plenty of stamps of the values 1d., 2d., 3d., 4d., and 5d., what is the greatest value that you can stick on the card if the Chancellor of the Exchequer forbids you to place any stamp in a straight line (that is, horizontally, vertically, or diagonally) with another stamp of similar value? Of course, only one stamp can be affixed in a s.p.a.ce. The reader will probably find, when he sees the solution, that, like the stamps themselves, he is licked He will most likely be twopence short of the maximum. A friend asked the Post Office how it was to be done; but they sent him to the Customs and Excise officer, who sent him to the Insurance Commissioners, who sent him to an approved society, who profanely sent him--but no matter.

309.--THE FORTY-NINE COUNTERS.

[Ill.u.s.tration]

Can you rearrange the above forty-nine counters in a square so that no letter, and also no number, shall be in line with a similar one, vertically, horizontally, or diagonally? Here I, of course, mean in the lines parallel with the diagonals, in the chessboard sense.

310.--THE THREE SHEEP.

[Ill.u.s.tration]

A farmer had three sheep and an arrangement of sixteen pens, divided off by hurdles in the manner indicated in the ill.u.s.tration. In how many different ways could he place those sheep, each in a separate pen, so that every pen should be either occupied or in line (horizontally, vertically, or diagonally) with at least one sheep? I have given one arrangement that fulfils the conditions. How many others can you find? Mere reversals and reflections must not be counted as different. The reader may regard the sheep as queens. The problem is then to place the three queens so that every square shall be either occupied or attacked by at least one queen--in the maximum number of different ways.

311.--THE FIVE DOGS PUZZLE.