Kitabı oku: «The Canterbury Puzzles, and Other Curious Problems», sayfa 7
THE PROFESSOR'S PUZZLES
"Why, here is the Professor!" exclaimed Grigsby. "We'll make him show us some new puzzles."
It was Christmas Eve, and the club was nearly deserted. Only Grigsby, Hawkhurst, and myself, of all the members, seemed to be detained in town over the season of mirth and mince-pies. The man, however, who had just entered was a welcome addition to our number. "The Professor of Puzzles," as we had nicknamed him, was very popular at the club, and when, as on the present occasion, things got a little slow, his arrival was a positive blessing.
He was a man of middle age, cheery and kind-hearted, but inclined to be cynical. He had all his life dabbled in puzzles, problems, and enigmas of every kind, and what the Professor didn't know about these matters was admittedly not worth knowing. His puzzles always had a charm of their own, and this was mainly because he was so happy in dishing them up in palatable form.
"You are the man of all others that we were hoping would drop in," said Hawkhurst. "Have you got anything new?"
"I have always something new," was the reply, uttered with feigned conceit—for the Professor was really a modest man—"I'm simply glutted with ideas."
"Where do you get all your notions?" I asked.
"Everywhere, anywhere, during all my waking moments. Indeed, two or three of my best puzzles have come to me in my dreams."
"Then all the good ideas are not used up?"
"Certainly not. And all the old puzzles are capable of improvement, embellishment, and extension. Take, for example, magic squares. These were constructed in India before the Christian era, and introduced into Europe about the fourteenth century, when they were supposed to possess certain magical properties that I am afraid they have since lost. Any child can arrange the numbers one to nine in a square that will add up fifteen in eight ways; but you will see it can be developed into quite a new problem if you use coins instead of numbers."
67.—The Coinage Puzzle
He made a rough diagram, and placed a crown and a florin in two of the divisions, as indicated in the illustration.
"Now," he continued, "place the fewest possible current English coins in the seven empty divisions, so that each of the three columns, three rows, and two diagonals shall add up fifteen shillings. Of course, no division may be without at least one coin, and no two divisions may contain the same value."
"But how can the coins affect the question?" asked Grigsby.
"That you will find out when you approach the solution."
"I shall do it with numbers first," said Hawkhurst, "and then substitute coins."
Five minutes later, however, he exclaimed, "Hang it all! I can't help getting the 2 in a corner. May the florin be moved from its present position?"
"Certainly not."
"Then I give it up."
But Grigsby and I decided that we would work at it another time, so the Professor showed Hawkhurst the solution privately, and then went on with his chat.
68.—The Postage Stamps Puzzles
"Now, instead of coins we'll substitute postage-stamps. Take ten current English stamps, nine of them being all of different values, and the tenth a duplicate. Stick two of them in one division and one in each of the others, so that the square shall this time add up ninepence in the eight directions as before."
"Here you are!" cried Grigsby, after he had been scribbling for a few minutes on the back of an envelope.
The Professor smiled indulgently.
"Are you sure that there is a current English postage-stamp of the value of threepence-halfpenny?"
"For the life of me, I don't know. Isn't there?"
"That's just like the Professor," put in Hawkhurst. "There never was such a 'tricky' man. You never know when you have got to the bottom of his puzzles. Just when you make sure you have found a solution, he trips you up over some little point you never thought of."
"When you have done that," said the Professor, "here is a much better one for you. Stick English postage stamps so that every three divisions in a line shall add up alike, using as many stamps as you choose, so long as they are all of different values. It is a hard nut."
69.—The Frogs and Tumblers
"What do you think of these?"
The Professor brought from his capacious pockets a number of frogs, snails, lizards, and other creatures of Japanese manufacture—very grotesque in form and brilliant in colour. While we were looking at them he asked the waiter to place sixty-four tumblers on the club table. When these had been brought and arranged in the form of a square, as shown in the illustration, he placed eight of the little green frogs on the glasses as shown.
"Now," he said, "you see these tumblers form eight horizontal and eight vertical lines, and if you look at them diagonally (both ways) there are twenty-six other lines. If you run your eye along all these forty-two lines, you will find no two frogs are anywhere in a line.
"The puzzle is this. Three of the frogs are supposed to jump from their present position to three vacant glasses, so that in their new relative positions still no two frogs shall be in a line. What are the jumps made?"
"I suppose–" began Hawkhurst.
"I know what you are going to ask," anticipated the Professor. "No; the frogs do not exchange positions, but each of the three jumps to a glass that was not previously occupied."
"But surely there must be scores of solutions?" I said.
"I shall be very glad if you can find them," replied the Professor with a dry smile. "I only know of one—or rather two, counting a reversal, which occurs in consequence of the position being symmetrical."
70.—Romeo and Juliet
For some time we tried to make these little reptiles perform the feat allotted to them, and failed. The Professor, however, would not give away his solution, but said he would instead introduce to us a little thing that is childishly simple when you have once seen it, but cannot be mastered by everybody at the very first attempt.
"Waiter!" he called again. "Just take away these glasses, please, and bring the chessboards."
"I hope to goodness," exclaimed Grigsby, "you are not going to show us some of those awful chess problems of yours. 'White to mate Black in 427 moves without moving his pieces.' 'The bishop rooks the king, and pawns his Giuoco Piano in half a jiff.'"
"No, it is not chess. You see these two snails. They are Romeo and Juliet. Juliet is on her balcony, waiting the arrival of her love; but Romeo has been dining, and forgets, for the life of him, the number of her house. The squares represent sixty-four houses, and the amorous swain visits every house once and only once before reaching his beloved. Now, make him do this with the fewest possible turnings. The snail can move up, down, and across the board and through the diagonals. Mark his track with this piece of chalk."
"Seems easy enough," said Grigsby, running the chalk along the squares. "Look! that does it."
"Yes," said the Professor: "Romeo has got there, it is true, and visited every square once, and only once; but you have made him turn nineteen times, and that is not doing the trick in the fewest turns possible."
Hawkhurst, curiously enough, hit on the solution at once, and the Professor remarked that this was just one of those puzzles that a person might solve at a glance or not master in six months.
71.—Romeo's Second Journey
"It was a sheer stroke of luck on your part, Hawkhurst," he added. "Here is a much easier puzzle, because it is capable of more systematic analysis; yet it may just happen that you will not do it in an hour. Put Romeo on a white square and make him crawl into every other white square once with the fewest possible turnings. This time a white square may be visited twice, but the snail must never pass a second time through the same corner of a square nor ever enter the black squares."
"May he leave the board for refreshments?" asked Grigsby.
"No; he is not allowed out until he has performed his feat."
72.—The Frogs who would a-wooing go
While we were vainly attempting to solve this puzzle, the Professor arranged on the table ten of the frogs in two rows, as they will be found in the illustration.
"That seems entertaining," I said. "What is it?"
"It is a little puzzle I made a year ago, and a favourite with the few people who have seen it. It is called 'The Frogs who would a-wooing go.' Four of them are supposed to go a-wooing, and after the four have each made a jump upon the table, they are in such a position that they form five straight rows with four frogs in every row."
"What's that?" asked Hawkhurst. "I think I can do that." A few minutes later he exclaimed, "How's this?"
"They form only four rows instead of five, and you have moved six of them," explained the Professor.
"Hawkhurst," said Grigsby severely, "you are a duffer. I see the solution at a glance. Here you are! These two jump on their comrades' backs."
"No, no," admonished the Professor; "that is not allowed. I distinctly said that the jumps were to be made upon the table. Sometimes it passes the wit of man so to word the conditions of a problem that the quibbler will not persuade himself that he has found a flaw through which the solution may be mastered by a child of five."
After we had been vainly puzzling with these batrachian lovers for some time, the Professor revealed his secret.
The Professor gathered up his Japanese reptiles and wished us good-night with the usual seasonable compliments. We three who remained had one more pipe together, and then also left for our respective homes. Each believes that the other two racked their brains over Christmas in the determined attempt to master the Professor's puzzles; but when we next met at the club we were all unanimous in declaring that those puzzles which we had failed to solve "we really had not had time to look at," while those we had mastered after an enormous amount of labour "we had seen at the first glance directly we got home."
MISCELLANEOUS PUZZLES
73.—The Game of Kayles
Nearly all of our most popular games are of very ancient origin, though in many cases they have been considerably developed and improved. Kayles—derived from the French word quilles—was a great favourite in the fourteenth century, and was undoubtedly the parent of our modern game of ninepins. Kayle-pins were not confined in those days to any particular number, and they were generally made of a conical shape and set up in a straight row.
At first they were knocked down by a club that was thrown at them from a distance, which at once suggests the origin of the pastime of "shying for cocoanuts" that is to-day so popular on Bank Holidays on Hampstead Heath and elsewhere. Then the players introduced balls, as an improvement on the club.
In the illustration we get a picture of some of our fourteenth-century ancestors playing at kayle-pins in this manner.
Now, I will introduce to my readers a new game of parlour kayle-pins, that can be played across the table without any preparation whatever. You simply place in a straight row thirteen dominoes, chess-pawns, draughtsmen, counters, coins, or beans—anything will do—all close together, and then remove the second one as shown in the picture.
It is assumed that the ancient players had become so expert that they could always knock down any single kayle-pin, or any two kayle-pins that stood close together. They therefore altered the game, and it was agreed that the player who knocked down the last pin was the winner.
Therefore, in playing our table-game, all you have to do is to knock down with your fingers, or take away, any single kayle-pin or two adjoining kayle-pins, playing alternately until one of the two players makes the last capture, and so wins. I think it will be found a fascinating little game, and I will show the secret of winning.
Remember that the second kayle-pin must be removed before you begin to play, and that if you knock down two at once those two must be close together, because in the real game the ball could not do more than this.
74.—The Broken Chessboard
There is a story of Prince Henry, son of William the Conqueror, afterwards Henry I., that is so frequently recorded in the old chronicles that it is doubtless authentic. The following version of the incident is taken from Hayward's Life of William the Conqueror, published in 1613—
"Towards the end of his reigne he appointed his two sonnes Robert and Henry, with joynt authoritie, governours of Normandie; the one to suppresse either the insolence or levitie of the other. These went together to visit the French king lying at Constance: where, entertaining the time with varietie of disports, Henry played with Louis, then Daulphine of France, at chesse, and did win of him very much.
"Hereat Louis beganne to growe warme in words, and was therein little respected by Henry. The great impatience of the one and the small forbearance of the other did strike in the end such a heat between them that Louis threw the chessmen at Henry's face.
"Henry again stroke Louis with the chessboard, drew blood with the blowe, and had presently slain him upon the place had he not been stayed by his brother Robert.
"Hereupon they presently went to horse, and their spurres claimed so good haste as they recovered Pontoise, albeit they were sharply pursued by the French."
Now, tradition—on this point not trustworthy—says that the chessboard broke into the thirteen fragments shown in our illustration. It will be seen that there are twelve pieces, all different in shape, each containing five squares, and one little piece of four squares only.
We thus have all the sixty-four squares of the chess-board, and the puzzle is simply to cut them out and fit them together, so as to make a perfect board properly chequered. The pieces may be easily cut out of a sheet of "squared" paper, and, if mounted on cardboard, they will form a source of perpetual amusement in the home.
If you succeed in constructing the chessboard, but do not record the arrangement, you will find it just as puzzling the next time you feel disposed to attack it.
Prince Henry himself, with all his skill and learning, would have found it an amusing pastime.
75.—The Spider and the Fly
Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.
76.—The Perplexed Cellarman
Here is a little puzzle culled from the traditions of an old monastery in the west of England. Abbot Francis, it seems, was a very worthy man; and his methods of equity extended to those little acts of charity for which he was noted for miles round.
The Abbot, moreover, had a fine taste in wines. On one occasion he sent for the cellarman, and complained that a particular bottling was not to his palate.
"Pray tell me, Brother John, how much of this wine thou didst bottle withal."
"A fair dozen in large bottles, my lord abbot, and the like in the small," replied the cellarman, "whereof five of each have been drunk in the refectory."
"So be it. There be three varlets waiting at the gate. Let the two dozen bottles be given unto them, both full and empty; and see that the dole be fairly made, so that no man receive more wine than another, nor any difference in bottles."
Poor John returned to his cellar, taking the three men with him, and then his task began to perplex him. Of full bottles he had seven large and seven small, and of empty bottles five large and five small, as shown in the illustration. How was he to make the required equitable division?
He divided the bottles into three groups in several ways that at first sight seemed to be quite fair, since two small bottles held just the same quantity of wine as one large one. But the large bottles themselves, when empty, were not worth two small ones.
Hence the abbot's order that each man must take away the same number of bottles of each size.
Finally, the cellarman had to consult one of the monks who was good at puzzles of this kind, and who showed him how the thing was done. Can you find out just how the distribution was made?
77.—Making a Flag
A good dissection puzzle in so few as two pieces is rather a rarity, so perhaps the reader will be interested in the following. The diagram represents a piece of bunting, and it is required to cut it into two pieces (without any waste) that will fit together and form a perfectly square flag, with the four roses symmetrically placed. This would be easy enough if it were not for the four roses, as we should merely have to cut from A to B, and insert the piece at the bottom of the flag. But we are not allowed to cut through any of the roses, and therein lies the difficulty of the puzzle. Of course we make no allowance for "turnings."
78.—Catching the Hogs
In the illustration Hendrick and Katrün are seen engaged in the exhilarating sport of attempting the capture of a couple of hogs.
Why did they fail?
Strange as it may seem, a complete answer is afforded in the little puzzle game that I will now explain.
Copy the simple diagram on a conveniently large sheet of cardboard or paper, and use four marked counters to represent the Dutchman, his wife, and the two hogs.
At the beginning of the game these must be placed on the squares on which they are shown. One player represents Hendrick and Katrün, and the other the hogs. The first player moves the Dutchman and his wife one square each in any direction (but not diagonally), and then the second player moves both pigs one square each (not diagonally); and so on, in turns, until Hendrick catches one hog and Katrün the other.
This you will find would be absurdly easy if the hogs moved first, but this is just what Dutch pigs will not do.
79.—The Thirty-one Game
This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.
As, on its own merits, however, the game is particularly interesting, I will make no apology for presenting it to my readers.
The cardsharper lays down the twenty-four cards shown in the illustration, and invites the innocent wayfarer to try his luck or skill by seeing which of them can first score thirty-one, or drive his opponent beyond, in the following manner:—
One player turns down a card, say a 2, and counts "two"; the second player turns down a card, say a 5, and, adding this to the score, counts "seven"; the first player turns down another card, say a 1, and counts "eight"; and so the play proceeds alternately until one of them scores the "thirty-one," and so wins.
Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play? The reader will perhaps say: "Oh, that is easy enough. You must play first, and turn down a 3; then, whatever your opponent does, he cannot stop your making ten, or stop your making seventeen, twenty-four, and the winning thirty-one. You have only to secure these numbers to win."
But this is just that little knowledge which is such a dangerous thing, and it places you in the hands of the sharper.
You play 3, and the sharper plays 4 and counts "seven"; you play 3 and count "ten"; the sharper turns down 3 and scores "thirteen"; you play 4 and count "seventeen"; the sharper plays a 4 and counts "twenty-one"; you play 3 and make your "twenty-four."
Now the sharper plays the last 4 and scores "twenty-eight." You look in vain for another 3 with which to win, for they are all turned down! So you are compelled either to let him make the "thirty-one" or to go yourself beyond, and so lose the game.
You thus see that your method of certainly winning breaks down utterly, by what may be called the "method of exhaustion." I will give the key to the game, showing how you may always win; but I will not here say whether you must play first or second: you may like to find it out for yourself.
