Amusements in Mathematics Part 12

In 1863, C.F. de Jaenisch first discussed the "Five Queens Puzzle"--to place five queens on the chessboard so that every square shall be attacked or occupied--which was propounded by his friend, a "Mr. de R." Jaenisch showed that if no queen may attack another there are ninety-one different ways of placing the five queens, reversals and reflections not counting as different. If the queens may attack one another, I have recorded hundreds of ways, but it is not practicable to enumerate them exactly.

[Ill.u.s.tration]

The ill.u.s.tration is supposed to represent an arrangement of sixty-four kennels. It will be seen that five kennels each contain a dog, and on further examination it will be seen that every one of the sixty-four kennels is in a straight line with at least one dog--either horizontally, vertically, or diagonally. Take any kennel you like, and you will find that you can draw a straight line to a dog in one or other of the three ways mentioned. The puzzle is to replace the five dogs and discover in just how many different ways they may be placed in five kennels in a straight row, so that every kennel shall always be in line with at least one dog. Reversals and reflections are here counted as different.

312.--THE FIVE CRESCENTS OF BYZANTIUM.

When Philip of Macedon, the father of Alexander the Great, found himself confronted with great difficulties in the siege of Byzantium, he set his men to undermine the walls. His desires, however, miscarried, for no sooner had the operations been begun than a crescent moon suddenly appeared in the heavens and discovered his plans to his adversaries. The Byzantines were naturally elated, and in order to show their grat.i.tude they erected a statue to Diana, and the crescent became thenceforward a symbol of the state. In the temple that contained the statue was a square pavement composed of sixty-four large and costly tiles. These were all plain, with the exception of five, which bore the symbol of the crescent. These five were for occult reasons so placed that every tile should be watched over by (that is, in a straight line, vertically, horizontally, or diagonally with) at least one of the crescents. The arrangement adopted by the Byzantine architect was as follows:-- [Ill.u.s.tration]

Now, to cover up one of these five crescents was a capital offence, the death being something very painful and lingering. But on a certain occasion of festivity it was necessary to lay down on this pavement a square carpet of the largest dimensions possible, and I have shown in the ill.u.s.tration by dark shading the largest dimensions that would be available.

The puzzle is to show how the architect, if he had foreseen this question of the carpet, might have so arranged his five crescent tiles in accordance with the required conditions, and yet have allowed for the largest possible square carpet to be laid down without any one of the five crescent tiles being covered, or any portion of them.

313.--QUEENS AND BISHOP PUZZLE.

It will be seen that every square of the board is either occupied or attacked. The puzzle is to subst.i.tute a bishop for the rook on the same square, and then place the four queens on other squares so that every square shall again be either occupied or attacked.

[Ill.u.s.tration]

314.--THE SOUTHERN CROSS.

[Ill.u.s.tration]

In the above ill.u.s.tration we have five Planets and eighty-one Fixed Stars, five of the latter being hidden by the Planets. It will be found that every Star, with the exception of the ten that have a black spot in their centres, is in a straight line, vertically, horizontally, or diagonally, with at least one of the Planets. The puzzle is so to rearrange the Planets that all the Stars shall be in line with one or more of them.

In rearranging the Planets, each of the five may be moved once in a straight line, in either of the three directions mentioned. They will, of course, obscure five other Stars in place of those at present covered.

315.--THE HAT-PEG PUZZLE.

Here is a five-queen puzzle that I gave in a fanciful dress in 1897. As the queens were there represented as hats on sixty-four pegs, I will keep to the t.i.tle, "The Hat-Peg Puzzle." It will be seen that every square is occupied or attacked. The puzzle is to remove one queen to a different square so that still every square is occupied or attacked, then move a second queen under a similar condition, then a third queen, and finally a fourth queen. After the fourth move every square must be attacked or occupied, but no queen must then attack another. Of course, the moves need not be "queen moves;" you can move a queen to any part of the board.

[Ill.u.s.tration]

316.--THE AMAZONS.

[Ill.u.s.tration]

This puzzle is based on one by Captain Turton. Remove three of the queens to other squares so that there shall be eleven squares on the board that are not attacked. The removal of the three queens need not be by "queen moves." You may take them up and place them anywhere. There is only one solution.

317.--A PUZZLE WITH p.a.w.nS.

Place two p.a.w.ns in the middle of the chessboard, one at Q 4 and the other at K 5. Now, place the remaining fourteen p.a.w.ns (sixteen in all) so that no three shall be in a straight line in any possible direction.

Note that I purposely do not say queens, because by the words "any possible direction" I go beyond attacks on diagonals. The p.a.w.ns must be regarded as mere points in s.p.a.ce--at the centres of the squares. See dotted lines in the case of No. 300, "The Eight Queens."

318.--LION-HUNTING.

[Ill.u.s.tration]

My friend Captain Potham Hall, the renowned hunter of big game, says there is nothing more exhilarating than a brush with a herd--a pack--a team--a flock--a swarm (it has taken me a full quarter of an hour to recall the right word, but I have it at last)--a pride of lions. Why a number of lions are called a "pride," a number of whales a "school," and a number of foxes a "skulk" are mysteries of philology into which I will not enter.

Well, the captain says that if a spirited lion crosses your path in the desert it becomes lively, for the lion has generally been looking for the man just as much as the man has sought the king of the forest. And yet when they meet they always quarrel and fight it out. A little contemplation of this unfortunate and long-standing feud between two estimable families has led me to figure out a few calculations as to the probability of the man and the lion crossing one another's path in the jungle. In all these cases one has to start on certain more or less arbitrary a.s.sumptions. That is why in the above ill.u.s.tration I have thought it necessary to represent the paths in the desert with such rigid regularity. Though the captain a.s.sures me that the tracks of the lions usually run much in this way, I have doubts.

The puzzle is simply to find out in how many different ways the man and the lion may be placed on two different spots that are not on the same path. By "paths" it must be understood that I only refer to the ruled lines. Thus, with the exception of the four corner spots, each combatant is always on two paths and no more. It will be seen that there is a lot of scope for evading one another in the desert, which is just what one has always understood.

319.--THE KNIGHT-GUARDS.

[Ill.u.s.tration]

The knight is the irresponsible low comedian of the chessboard. "He is a very uncertain, sneaking, and demoralizing rascal," says an American writer. "He can only move two squares, but makes up in the quality of his locomotion for its quant.i.ty, for he can spring one square sideways and one forward simultaneously, like a cat; can stand on one leg in the middle of the board and jump to any one of eight squares he chooses; can get on one side of a fence and blackguard three or four men on the other; has an objectionable way of inserting himself in safe places where he can scare the king and compel him to move, and then gobble a queen. For pure cussedness the knight has no equal, and when you chase him out of one hole he skips into another." Attempts have been made over and over again to obtain a short, simple, and exact definition of the move of the knight--without success. It really consists in moving one square like a rook, and then another square like a bishop--the two operations being done in one leap, so that it does not matter whether the first square pa.s.sed over is occupied by another piece or not. It is, in fact, the only leaping move in chess. But difficult as it is to define, a child can learn it by inspection in a few minutes.

I have shown in the diagram how twelve knights (the fewest possible that will perform the feat) may be placed on the chessboard so that every square is either occupied or attacked by a knight. Examine every square in turn, and you will find that this is so. Now, the puzzle in this case is to discover what is the smallest possible number of knights that is required in order that every square shall be either occupied or attacked, and every knight protected by another knight. And how would you arrange them? It will be found that of the twelve shown in the diagram only four are thus protected by being a knight's move from another knight.

THE GUARDED CHESSBOARD.

On an ordinary chessboard, 8 by 8, every square can be guarded--that is, either occupied or attacked--by 5 queens, the fewest possible. There are exactly 91 fundamentally different arrangements in which no queen attacks another queen. If every queen must attack (or be protected by) another queen, there are at fewest 41 arrangements, and I have recorded some 150 ways in which some of the queens are attacked and some not, but this last case is very difficult to enumerate exactly.

On an ordinary chessboard every square can be guarded by 8 rooks (the fewest possible) in 40,320 ways, if no rook may attack another rook, but it is not known how many of these are fundamentally different. (See solution to No. 295, "The Eight Rooks.") I have not enumerated the ways in which every rook shall be protected by another rook.

On an ordinary chessboard every square can be guarded by 8 bishops (the fewest possible), if no bishop may attack another bishop. Ten bishops are necessary if every bishop is to be protected. (See Nos. 297 and 298, "Bishops unguarded" and "Bishops guarded.") On an ordinary chessboard every square can be guarded by 12 knights if all but 4 are unprotected. But if every knight must be protected, 14 are necessary. (See No. 319, "The Knight-Guards.") Dealing with the queen on n boards generally, where n is less than 8, the following results will be of interest:-- 1 queen guards 2 board in 1 fundamental way.

1 queen guards 3 board in 1 fundamental way.

2 queens guard 4 board in 3 fundamental ways (protected).

3 queens guard 4 board in 2 fundamental ways (not protected).

3 queens guard 5 board in 37 fundamental ways (protected).

3 queens guard 5 board in 2 fundamental ways (not protected).

3 queens guard 6 board in 1 fundamental way (protected).

4 queens guard 6 board in 17 fundamental ways (not protected).

4 queens guard 7 board in 5 fundamental ways (protected).

4 queens guard 7 board in 1 fundamental way (not protected).

NON-ATTACKING CHESSBOARD ARRANGEMENTS.

We know that n queens may always be placed on a square board of n squares (if n be greater than 3) without any queen attacking another queen. But no general formula for enumerating the number of different ways in which it may be done has yet been discovered; probably it is undiscoverable. The known results are as follows:-- Where n = 4 there is 1 fundamental solution and 2 in all.

Where n = 5 there are 2 fundamental solutions and 10 in all.

Where n = 6 there is 1 fundamental solution and 4 in all.

Where n = 7 there are 6 fundamental solutions and 40 in all.

Where n = 8 there are 12 fundamental solutions and 92 in all.

Where n = 9 there are 46 fundamental solutions.

Where n = 10 there are 92 fundamental solutions.

Where n = 11 there are 341 fundamental solutions.

Obviously n rooks may be placed without attack on an n board in n! ways, but how many of these are fundamentally different I have only worked out in the four cases where n equals 2, 3, 4, and 5. The answers here are respectively 1, 2, 7, and 23. (See No. 296, "The Four Lions.") We can place 2n-2 bishops on an n board in 2^{n} ways. (See No. 299, "Bishops in Convocation.") For boards containing 2, 3, 4, 5, 6, 7, 8 squares, on a side there are respectively 1, 2, 3, 6, 10, 20, 36 fundamentally different arrangements. Where n is odd there are 2^{(n-1)} such arrangements, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{(n-3)} giving 8. Where n is even there are 2^{(n-2)}, each giving 4 by reversals and reflections, and 2^{n-3} - 2^{(n-4)}, each giving 8.

We can place (n+1) knights on an n board without attack, when n is odd, in 1 fundamental way; and n knights on an n board, when n is even, in 1 fundamental way. In the first case we place all the knights on the same colour as the central square; in the second case we place them all on black, or all on white, squares.

THE TWO PIECES PROBLEM.

On a board of n squares, two queens, two rooks, two bishops, or two knights can always be placed, irrespective of attack or not, in (n^{4} - n) ways. The following formulae will show in how many of these ways the two pieces may be placed with attack and without:-- With Attack. Without Attack.

2 Queens 5n - 6n + n 3n^{4} - 10n + 9n - 2n ------------------- ------------------------------ 3 6 2 Rooks n - n n^{4} - 2n + n ---------------------- 2 2 Bishops 4n - 6n + 2n 3n^{4} - 4n + 3n - 2n -------------------- ----------------------------- 6 6 2 Knights 4n - 12n + 8 n^{4} - 9n + 24n -------------------- 2 (See No. 318, " Lion Hunting.") DYNAMICAL CHESS PUZZLES.

"Push on--keep moving." THOS. MORTON: Cure for the Heartache.

320.--THE ROOK'S TOUR.

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

The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pa.s.s over it or make it a stopping-place, and we will not quibble over the point whether the original square is actually visited twice. We will a.s.sume that it is not.

321.--THE ROOK'S JOURNEY.

This puzzle I call "The Rook's Journey," because the word "tour" (derived from a turner's wheel) implies that we return to the point from which we set out, and we do not do this in the present case. We should not be satisfied with a personally conducted holiday tour that ended by leaving us, say, in the middle of the Sahara. The rook here makes twenty-one moves, in the course of which journey it visits every square of the board once and only once, stopping at the square marked 10 at the end of its tenth move, and ending at the square marked 21. Two consecutive moves cannot be made in the same direction--that is to say, you must make a turn after every move.

[Ill.u.s.tration: +---+---+---+---+---+---+---+---+ | | | | | | | | R | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | | | | | | | | +---+---+---+---+---+---+---+---+ | | 21| | 10| | | | | +---+---+---+---+---+---+---+---+ ]

322.--THE LANGUISHING MAIDEN.

[Ill.u.s.tration: --+-----+-----+-----+-----+-----+-----+-----+ | | | | | | | | | Kt | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | M | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ | | | | | | | | | | | | | | | | | | | | +-- --+-- --+-- --+-- --+-- --+-- --+-- --+-- --+ ]

A wicked baron in the good old days imprisoned an innocent maiden in one of the deepest dungeons beneath the castle moat. It will be seen from our ill.u.s.tration that there were sixty-three cells in the dungeon, all connected by open doors, and the maiden was chained in the cell in which she is shown. Now, a valiant knight, who loved the damsel, succeeded in rescuing her from the enemy. Having gained an entrance to the dungeon at the point where he is seen, he succeeded in reaching the maiden after entering every cell once and only once. Take your pencil and try to trace out such a route. When you have succeeded, then try to discover a route in twenty-two straight paths through the cells. It can be done in this number without entering any cell a second time.

323.--A DUNGEON PUZZLE.

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

A French prisoner, for his sins (or other people's), was confined in an underground dungeon containing sixty-four cells, all communicating with open doorways, as shown in our ill.u.s.tration. In order to reduce the tedium of his restricted life, he set himself various puzzles, and this is one of them. Starting from the cell in which he is shown, how could he visit every cell once, and only once, and make as many turnings as possible? His first attempt is shown by the dotted track. It will be found that there are as many as fifty-five straight lines in his path, but after many attempts he improved upon this. Can you get more than fifty-five? You may end your path in any cell you like. Try the puzzle with a pencil on chessboard diagrams, or you may regard them as rooks' moves on a board.

324.--THE LION AND THE MAN.

In a public place in Rome there once stood a prison divided into sixty-four cells, all open to the sky and all communicating with one another, as shown in the ill.u.s.tration. The sports that here took place were watched from a high tower. The favourite game was to place a Christian in one corner cell and a lion in the diagonally opposite corner and then leave them with all the inner doors open. The consequent effect was sometimes most laughable. On one occasion the man was given a sword. He was no coward, and was as anxious to find the lion as the lion undoubtedly was to find him.

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

The man visited every cell once and only once in the fewest possible straight lines until he reached the lion's cell. The lion, curiously enough, also visited every cell once and only once in the fewest possible straight lines until he finally reached the man's cell. They started together and went at the same speed; yet, although they occasionally got glimpses of one another, they never once met. The puzzle is to show the route that each happened to take.

325.--AN EPISCOPAL VISITATION.

The white squares on the chessboard represent the parishes of a diocese. Place the bishop on any square you like, and so contrive that (using the ordinary bishop's move of chess) he shall visit every one of his parishes in the fewest possible moves. Of course, all the parishes pa.s.sed through on any move are regarded as "visited." You can visit any squares more than once, but you are not allowed to move twice between the same two adjoining squares. What are the fewest possible moves? The bishop need not end his visitation at the parish from which he first set out.

326.--A NEW COUNTER PUZZLE.

Here is a new puzzle with moving counters, or coins, that at first glance looks as if it must be absurdly simple. But it will be found quite a little perplexity. I give it in this place for a reason that I will explain when we come to the next puzzle. Copy the simple diagram, enlarged, on a sheet of paper; then place two white counters on the points 1 and 2, and two red counters on 9 and 10, The puzzle is to make the red and white change places. You may move the counters one at a time in any order you like, along the lines from point to point, with the only restriction that a red and a white counter may never stand at once on the same straight line. Thus the first move can only be from 1 or 2 to 3, or from 9 or 10 to 7.

[Ill.u.s.tration: 4 8 / / 2 6 10 / / 3 7 / / 1 5 9 ]

327.--A NEW BISHOP'S PUZZLE.

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

This is quite a fascinating little puzzle. Place eight bishops (four black and four white) on the reduced chessboard, as shown in the ill.u.s.tration. The problem is to make the black bishops change places with the white ones, no bishop ever attacking another of the opposite colour. They must move alternately--first a white, then a black, then a white, and so on. When you have succeeded in doing it at all, try to find the fewest possible moves.

If you leave out the bishops standing on black squares, and only play on the white squares, you will discover my last puzzle turned on its side.

328.--THE QUEEN'S TOUR.

The puzzle of making a complete tour of the chessboard with the queen in the fewest possible moves (in which squares may be visited more than once) was first given by the late Sam Loyd in his Chess Strategy. But the solution shown below is the one he gave in _American Chess-Nuts_ in 1868. I have recorded at least six different solutions in the minimum number of moves--fourteen--but this one is the best of all, for reasons I will explain.

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

If you will look at the lettered square you will understand that there are only ten really differently placed squares on a chessboard--those enclosed by a dark line--all the others are mere reversals or reflections. For example, every A is a corner square, and every J a central square. Consequently, as the solution shown has a turning-point at the enclosed D square, we can obtain a solution starting from and ending at any square marked D--by just turning the board about. Now, this scheme will give you a tour starting from any A, B, C, D, E, F, or H, while no other route that I know can be adapted to more than five different starting-points. There is no Queen's Tour in fourteen moves (remember a tour must be re-entrant) that may start from a G, I, or J. But we can have a non-re-entrant path over the whole board in fourteen moves, starting from any given square. Hence the following puzzle:-- [Ill.u.s.tration: +---+---+---+---*---+---+---+---+ | A | B | C | G " G | C | B | A | *===*---+---+---*---+---+---+---+ | B " D | E | H " H | E | D | B | +---*===*---+---*---+---+---+---+ | C | E " F | I " I | F | E | C | +---+---*===*---*---+---+---+---+ | G | H | I " J " J | I | H | G | +---+---+---*===*---+---+---+---+ | G | H | I | J | J | I | H | G | +---+---+---+---+---+---+---+---+ | C | E | F | I | I | F | E | C | +---+---+---+---+---+---+---+---+ | B | D | E | H | H | E | D | B | +---+---+---+---+---+---+---+---+ | A | B | C | G | G | C | B | A | +---+---+---+---+---+---+---+---+ ]

Start from the J in the enclosed part of the lettered diagram and visit every square of the board in fourteen moves, ending wherever you like.

329.--THE STAR PUZZLE.

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

Put the point of your pencil on one of the white stars and (without ever lifting your pencil from the paper) strike out all the stars in fourteen continuous straight strokes, ending at the second white star. Your straight strokes may be in any direction you like, only every turning must be made on a star. There is no objection to striking out any star more than once.

In this case, where both your starting and ending squares are fixed inconveniently, you cannot obtain a solution by breaking a Queen's Tour, or in any other way by queen moves alone. But you are allowed to use oblique straight lines--such as from the upper white star direct to a corner star.

330.--THE YACHT RACE.

Now then, ye land-lubbers, hoist your baby-jib-topsails, break out your spinnakers, ease off your balloon sheets, and get your head-sails set!

Our race consists in starting from the point at which the yacht is lying in the ill.u.s.tration and touching every one of the sixty-four buoys in fourteen straight courses, returning in the final tack to the buoy from which we start. The seventh course must finish at the buoy from which a flag is flying.

This puzzle will call for a lot of skilful seamanship on account of the sharp angles at which it will occasionally be necessary to tack. The point of a lead pencil and a good nautical eye are all the outfit that we require.

[Ill.u.s.tration]

This is difficult, because of the condition as to the flag-buoy, and because it is a re-entrant tour. But again we are allowed those oblique lines.