INTELLECT SHARPENERS.

Otago Daily Times, Issue 19970, 11 December 1926, Page 26

INTELLECT SHARPENERS.

Special to the Otago Daily Times. By T. L. Beiton. Readers with a little ingenuity will find in this column an abundant store of entertainment and amusement, and the solving of the problems should provide excellent mental exhilaration. While some of the “nuts” may appear harder than others, it will be found that none will require a sledgehammer to crack them. SQUARES AND CIRCLES. Many readers will recollect the large rewards that were offered by tobacco manufacturers and others to anyone who invented for their use a cutting contrivance that would open a square tin hermetically sealed, with the same facility, and by a similarly simple process, as the patented cutter we see attached to round tins containing wellknown brands of tobacco and cigarettes. Of course,, the manufacturers saw in such an invention the enormous savings that could be effected in freights if they could pack . their goods in square tins instead of round I ones, if at the same time the tins could be made airtight and capable of being opened by the <• -asumer without any difficulty. There is a little problem suggested by this. • Take a sheet of paper of any small size, say. Sin x Sin. We know that it is possible to place flat on this sheet 15 square counters each one inch in diameter, without overlapping eit-.er themselves or the edges of the paper. What is the greatest number of circular counters, each one inch in diameter, that could be similarly so placed, and what area of the paper would be left uncovered, and. so to speak, wasted? It is useful to known that a half-penny is exactly one inch in diameter. PATIENCE. 1 was spending the evening with some friends at South Dunedin recently/ and the subject of Christmas problems and puzzles was being discussed theoretically and in practice. The eldc't child, a bright boy of about 12, asked- me why the editor of the Daily Times did not put in some “nuts” not quite so hard as the Intellect Sharpeners. So I promised to mention the matter to the editor, with the result that he has agreed to put in the 'jllowlng problem for boys (and girls too) who are not quite grown up, but are fast growing that way. Incidentally, it is as good as a game of patience for much older folks than our young friend. Draw a six square figure on a sheet of paper, letter the spaces, and place counters, on the | A \ — ~j X j squares similarly let- | B |~~C | Y j tered. The top centre square is vacant. The problem is to place the piece on Y to X’s square, and X to Y’s square in the fewest possible moves. _ The counters may be moved one at a time to any vacant square, horizontally or perpendicularly only. No jumping over or taking pieces is allowed. It is a capital puzzle, and if the reader can do it in 16 moves he will accomplish the feat in one less than the composer can. MATHEMATICAL BOMBSHELL. To prove that two equals are one—for example, that two shillings are equal to only one of them in intrinsic value, seems a formidable proposition. Still, there is said to be quite a number of people to whom the mathematical proof of it is acceptable, some of whom it is rumoured have expressed the opinion “that very soon we will be told that two and two do not make four.” I merely mention this without comment, beyond stating that I do not yet see any cause for alarm in this respect. But a friend told me the other day when discussing the matter, that a young Timaru student who is at present up to conic sections, remarked, upon being shown the extraordinary proof of this problem, that he was not prepared to question the mathematical accuracy of it. However that may be, the problem is not for academic criticism, though this much may be said, that the steps in the working of the solution are beyond cavil on the part of the mathematician. CHANGE. Mr Smith told me of an incident that occurred to him this morning that is rather interesting for problem purposes. A friend asked him at the railway , station if he could change a ten-shilling note. As he had a few minutes before handed a pound note at the window when purchasing a short-distance ticket, receiving the change all in silver, he thought he could oblige his friend. He found, however, that although he had much more than ten shillings in silver, he could not give the e ? ca ®‘' change. He told me the amount he had, and upon examining the incident further. I found that the sum was the largest amount possible to have in current New Zealand silver coin without being able to change a ten-shilling note. What wa’s the sum that Mr Smith had 1 BY THE CLOCK. I don’t know whether my friend Hopkins thinks that problems are the begining and end of my daily existence, but if ever I chance to seek any information from him—which I always try now to avoid —he invariably responds in some enigmatical form or other. For instance, we had decided to go to one of the theatres the other night, and, after dinner, as we had not reserved our seats, I said we had, perhaps, better go a little earlier. “What is the time now? 1 asked. Hopkins looked at his watch thoughtfully, and said: “Exactly an hour, and twenty-one minutes ago it was twice as many minutes past 6 o’clock as it is 8 now; we have plenty of time. So we strolled leisurely down, and arrived at the theatre as the clock struck 8. What time was it when Hopkins looked at his watch! SOLUTION OF LAST WEEK’S PROBLEMS. A FAMILY CONUNDRUM. Henry, the nephew of Charles, was the son of Mavis, who had married a man of the same name as herself. A FASCINATING PUZZLE GAME. With 27 counters, the first player drawing 1,2, 3, or 4at a time, shouldwin every time. But he must take either 3 or 4 at his first play, and then, when lie is holding an odd number, he should leave in the heap one more than a multiple of 6; and, when holding even, he should leave one less than a multiple of six. This applies to any number of counters that is three more than u multiple of six, and the same rule applies when the number is only one more than a multiple of six, but in that case the player should take only 1 or 2 at his first play. If, however, the number of counters be one less than a multiple of six, as with 23, he should allow his opponent to play first, and, by observing the above rules, he will always win. THE BOYS’ MARBLES. The boys commenced their game with 100 marbles each, TWO SQUARES. The returned soldier, having some knowledge of mathematics, ran the four lines of his fencing at right angles to the diagonals of the original square paddock, leaving the four totara trees a few inches outside his boundaries. His enclosure was then in the form of a perfect square which included approximately 200 acres, just double what the sheep owner intended to give, yet fenced in strict accordance with the conditions. THE NINE COUNTERS. The object may be achieved in five moves, which will leave the ninth counter the only occupant of the board with an uninterrupted passage to the central square. Readers are requested not to send in their solutions, unless there are specially asked for, but to keep them for comparison with those published on the Saturday following the publication of the problems.