Posers for Puzzle Cranks

Taranaki Daily News, 17 December 1926, Page 8 (Supplement)

Posers for Puzzle Cranks

GAMES AND DEVICES FOR | CHRISTMAS

If a stranger had dropped in at the Cranks’ Club in Piccadilly on Christmas Eve, and glanced at the four men happily grouped round the smokingroom fire, he would probably have said, “Haven’t they any homes?” For Christmas, of all times of the year, is universally devoted to family home gatherings. But it must not be imagined that these men were without family ties, or that they had any domestic troubles, or even that they were unsociable. They just happened to be stranded in London at this time from various causes, for these accidents will happen in the best regulated families. Hut their reasons for spending the night at their Club do not really concern us. Let me just introduce them. Harold Musgrave is a young barrister, not altogether briefless, and he is an excellent mathematician. Gerald Marks is “something in the City”—l think he is a ship-broker, whose business, I understand, is not the breaking up of ships, but has to do with such things as charterparties, freights, and bills of lading. Tom Winterhouse Is —well, I hardly think he could tell you himself what he is, for he has ample private means, and is always talking about “being about to do” certain things, which have a way of never being done. He is a good-hearted fellow, but his best friend or his worst enemy cannot accuse him of being particularly clever. The fourth man is Guy Kilburn. He is a little older than the others, and is, I believe, a retired accountant, with, consequently, a considerable aptitude for figures. These men are all fond of a good puzzle. In fact, that is why we find them at the Cranks’ instead of at other Clubs, for the Club was devoted to the discussion of all kinds of queer problems, fallacies, and cranky theories. FOLDING A PENTAGON. It was Marks who started the ball rollling that evening by producing from his pocket a number of narrow ribbons of paper. He had asked them to form a regular pentagon with one of these strips without using pencil,

compasses, scissors, or anything else but the fingers. The solution he showed them is indicated in our fig. 1. If you tie the ribbon into a simple ordinary knot, and press flat and fold back at the dotted lines, you will have a regular pentagon obtained with .very little trouble. “Very interesting,” said Musgrave, “but can you do this: Take one of the ribbons of any length—say, more than four times as long as broad—and fold it, without the knot, into a perfect pentagon with every part lying within the boundaries of the figure? The angle ABC must be first cut for you at t!*6 correct angle for two contiguous sides of a regular pentagon. How are you going to fold it?” (See our fig. 2.

They all agreed that this was very easy, though elegant and instructive. “I have just been asked outside,” said Winterhouse, “to prove that a lazy dog is the same as a sheet of ruled foolscap paper. I can see no relation between the two things, but this is how my friend put it: ‘Well, you see, a lazy dog is the same as a slow pup, and we all know that a slope up is an Inclined plane. Then it is certain that an ink-lined plane may be a sheet of ruled foolscap paper —and there you are.’ ” They all smiled at this amusing little play upon words.

TRANSFERRING THE COUNTERS. Kilburn then laid out on the little table between them six playing cards, and placed on one of them, as shown in our illustration, a pile of fifteen coun-

ters which he had numbered consecutively, 1,2, 3, .... 15 downwards. "Now,” he said, “how are we to transfer the pile of counters in the fewest possible moves from card A to card F? You may move the counters one at a time to any card, but may never place a counter on one that bears a smaller number than itself. Thus, you see, if you place 1 on B and 2 on C, you can then place 1 on 2, but not 2 on 1.” This kept them amused for some littie time, for they had to convince one another that they, had found a solution in the fewest possible moves. "Before I forget it.” said Marks, “I wonder whether Winterhouse can answer this? A man had twentysix sheep and one died. How many remained?” “That is easy,” said Winterhouse, "it would be twenty-six if the dead one remained with the others.” "You are quite wrong, my dear fellow,” said Marks, “for under your conditions twenty would remain.” This they could none of them see un it he explained that he had not s-id "twenty-six sheep,” but "twenty sick shr- p ’•

LAMP SIGNALS. “But here is something,” said Musgrave, “in a little more serious vein. Two spies on the opposite sides of s river contrived a method for signalling at night. They each put up a stand like our illustration, and each possessed three signal lamps which couli show either white, red, or green light. They Constructed a • code in which every different signal meant a sen-

tence. You will, of course, sec thkt a single lamp hung on any one of the hooks could only mean the same thing, that two lamps hung on the upper hooks 1 and 2 could not be distinguished from two on 4 and 5. Of course, two red lamps on 1 and 5 could be distinguished from two on 1 and 6; and two on 1 and 2 would be different from two on 1 and 3. Now, remembering the variations of colour as well as of position, what is the greatest number of signals that could be sent?” After some little figuring and discussion they agreed on the correct solution. Then Winterhouse broke in with this little story: “You fellows all know Dick Wadhurst—that chap who does something in Somerset House—and you have experienced his love of a practical joke. But he came a cropper last week. After examining the dates on some coins he said to his' wife, ‘Do you know, my dear, that' 1900 pennies are now worm more than 7 guineas?’ She was much impressed, and after her husband had gone to his office she examined every penny she could come across in the search for those bearing the date 1900 and she was elated with her success. When Dick returned at night he found her trying on a hat of the latest Paris model, wearing a beautiful new coat, and surrounded by other similar luxuries. ‘Hullo!’ said Dick, ‘has some rich relation died and left you a fortune, dear, or what?’ She then explained the luck she had had that day having found no fewer than seven pennies of the date 1900, which were worth over £5O. ‘Stupid woman!’ said Wadhurst, in a fury. ‘Haven’t you enough brains to understand tiiat nineteen hundrrd pennies are always worth more than seven guineas whatever their date ?’ ” They all roared with laughter, and agreed that it served Dick right. It was well that he should sometimes be caught in his own trap. TURNING THE DIE. “Here is a little puzzle game,” said Marks, laying a die on the table, “that I think will interest you. The first player calls any number he chooses from 1 to 6, and the second player throws the die at hazard. Then we take it in turns to roll over the die in any direction we choose, but never giving it more than a quarter turn. The score increases as we proceed, and the player wins who manages to score 25 or forces his opponent to score beyond 25. Let us have a trial game, Kilburn.” Marks called 6 and Kilburn happened to throw a 3 (as shown by our illustration), making a score of 9.

Now Marks decided to turn up 1. scoring 10; Kilburn turned up 3, scoring 13; Marks turned up 6, scoring 19; Kilburn turned up 3, scoring 22; Marks turned up 1, scoring 23; and Kilburn turned up 2, scoring 25 and winning. What call should Marks have made in order to have the best chance of winning? Remember that the numbers on the opposite sides of a correct die always sum to 7, thus: I—6, 2—5, 3—4. A SWITCH PUZZLE. Here is a four-armed box, somewhat resembling a railway switch, containing twelve movable blocks, and the long arm will just hold nine blocks and the short arms two blocks with

room for one at the crossing. Musgrave produced a larger copy of this diagram, on cardboard, with twelve white counters, which he placed in the positions of the little white' squares shown. “Now,” he said, “select a word of twelve letters, and place one letter on each of the counters so that the word will read correctly from left to right. Then, in the fewest possible moves, slide the blocks Into the other arms of the box so that the word will read from the top to the bottom of the box. You will understand a move, is a slide of one letter any distance, whether you turn a corner or not. Lou must not lift the blocks from the box (or the counters from the diagram), so there arc no leaping moves.” “Of course,” said Winterhouse, “to do it in the fewest possible moves requires tlie selection of a favourable word.” The others of course agreed, and-l

after much trial they got very near to the correct answer, but, as they had toot hit on the best word possible, they made a few moves too many.

AT THE NORTH POLE.

“When I was lying awake In bed the other night,” said Winterhouse, “I imagined a.man going to the North Pole. The points of the compass N are, as everybody knows, W e. He S reached the Pole, and, having stepped over it, and passed on, must turn about to look North. But East is now on his left-hand side, West on his right-hand side, and the points of N the compass, therefore, are E W. Why S is this? Surely it is not possible ti alter the points of the compass by merely passing over the North Pole?” “You shouldn’t tax your brain, Winterhouse, with such difficult problems,” said Kilburn, and they all laughed. Then they explained the little difficulty to him, and, doubtless, it will not perplex the reader very much.

DIGITS AND PRIMES.

“Using all the nine digits once and once only,” said Musgrave, “can you find prime numbers that will add up to the smallest total possible?” Winterhouse seemed a little hazy as to the nature of a prime number, so it was explained that a prime is a number that cannot be divided without remainder by any number except one and itself, such as 1,3, 5,7, 11, 13, etc. This was Kilburn's first attempt : 61 283 47 59 450 It will be seen that the four numbers in the addition contain all the nine digits once and once only, and are all primes. But this total can be considerably reduced, and it is quite an easy puzzle if you use a little thought, LINES AND SQUARES. “With how few straight lines can you make exactly 100 squares?” asked (Marks, and he drew two diagrams like (those we give. “You see in this first

diagram it will be found that with nine straight lines I have made twenty squares, twelve with sides of the length AB, six with sides of the length AG, and two with sides of the length AD. In the second diagram, although I use one more line, I onlyget seventeen squares. So, you see, everything depends on how the lines are drawn.” “May I produce more than the required 100 squares?” asked Kilburn. “No,” Marks explained; “there must be exactly 100 squares, neither more nor fewer.” After they had solved this puzzle the hour was late, and they prepared to go to their respective homes, but, as they were wrapping up in the cloak room, Guy Kilburn said: “Here Is a little thing for you each to work on your way home.” THE MONEY PUZZLE. “A merchant noticed the curious fact that when he doubled £6 13s it became £l3 6s, thus merely exchanging the pounds and the shillings. He told me he had tried to discover another amount of money that had the same peculiarity, using any multiplier whatever, but he failed.” “Is there another number?” asked Musgrave. “Yes,” Kilburn assured them, “there is one and one only, and the multiplier is quite a small one, consisting of one figure only.” SOLUTIONS. FOLDING THE PENTAGON. By folding A over, find C so that BC equals AB. Then fold as in Fig. J across the point A, and this will

give you the point D. Now fold as in Fig. 2, making the edge of the ribbon lie along AB, and you will have the point E. Continue to fold, as in Fig. 3, and so on, until all the ribbon lies on the pentagon. TRANSFERRING THE COUNTERS. Make a pile of five counters (1 —5) on Bin nine moves. Make a pile of 4 (6—9) on C in seven moves. Make a pile of 3 (10 —12) on D in five moves. Make a pile of 2 (13 —14) on Ein three move. Place 1 (15) on Fin one move. Replace 13 and 14 on F in three, lO—l2 on F in five, 6—-9 in seven, and I—s1 —5 in nine moves. Forty-nine moves in all—the fewest possible. LAMP SIGNALS. With 3 red lamps or 3 white lamps or 3 green lamps we can make 151 variation each (45). With 1 red and 2 white we can make the same 15, and each way will admit of 3 variations of colour order, 45 in all. This is the same with 1 red and 2 green, 1 white and 2 red, 1 white and 2 green, 1 green and 2 white, and 1 green and 2 red (270). With 1 red, i whits and 1 green we can get 6 by 15 variations (90). With 2 red, or 2 white or 2 green we can get 7 patterns (21).| With 1 red and 1 white, or 1 red arid 1 green, or 1 white and 1 green we can get 14 variations each (42). With one lamp only we can only get one signal each (3). Add together the numbers in brackets (45, 270, 90, 21, 42, and 3), and we get the correct answer—47l ways.

TURNING THE DTE. The best call for the. first player Is either “two” or “three,” and in either case only one particular throw should defeat him. If he calls “one,” the throw of either three or six should Zlr.fsat !-; r . ;f ca ]] a “two,” ‘lie throw of five only should defeat him. Ii he calls “taive,” the throw of four only should defeat him. It' he calls “four,” the throw of either three or (four should defeat him. If he calls five,” the throw of either two or three should rtcl'cat him. If he calls “six,” the throw of cither one or five should defeat him. It is impossible here to give a complete analysis of the play, but I will just state that if at any time you score either 5,6, 9, 10, 14, 15, 18, 19, or 23, with the die any side up you ought to lose. If you score 7 or 16 with any side up you should win. The chance of winning with the other scores depends on the lie of the die. AT THE NORTH POLE. If W and E were stationary points, and W, as at present, on your left when advancing towards the North, then, after passing the Pole and turntog round, West would be on your right as stated. But W and E are not fixed points, but directions round the globe; so wherever you stand facing North you will have W direction on your left and the E direction on your right.

THE SWITCH PUZZLE. The solution in the fewest possible moves depends entirely on the selection of the most accommodating word. (Apart from the word condition, the (blocks themselves can be changed from horizontal to vertical order in 12 moves—the fewest possible. (Therefore, when we find by writing In the word “interpreting” we can do ‘what is required in 12 moves, we know that that solution cannot be (beaten. The moves are so obvious that I need not show them. I do not (know of any other word that lends 'itself to a solution in 12 moves. DIGITS AND PRIMES. The 4, 6 and 8 must come in the 10’s place, as no prime number can end with one of these, and 2 and 5 can only appear in the units place if alone. When these facts are noted, the rest is easy, as here shown: — 47 61 89 2 207

squares. . There are 40 with sides of t.ie length AB, 28 of the length AG, 18 of the length AD, 10 of the length AS, and 4 squares with sides of the length AF, making 100 in ail. It is possible with 15 straight lines to form 112. squares, but we were restricted to 100. With 14 straight h'nes you cannot form more than 91 squares. THE MONEY PUZZLE. The only other amount is £2 17s, which, multiplied by 6, produces £l7 2s—a mere exchange of the pounds and shillings.