NUTS TO CRACK

Otago Witness, Issue 4085, 28 June 1932, Page 8

NUTS TO CRACK

By

T. L. Briton.

(For thb Otago Witness.) Readers with a little ingenuity will - Ond In this column an abundant store of entertainment and amusement, and the solving of the problems should provide excellent mental exhilaration. While some of th. " nuts " may appear harder than others, it will be found that none will require a sledge-hammer to crack them. Solutions will appear In our ue.it Issue, together with some fresh “ nuts.’’ Readers are requested not .to send In their solutions unless these are specially asked for. but to keep them for comparison with those published in the Issue following the publication of the problems EXPRESSING “ 100.” M. Lucas, a French mathematician of note, stated in one of his several publications that there is no way of expressing 100 in the form of a. mixed number having only one figure in the integral part of it, and using each of the nine digits once and once only. It was left to the well-known mathematician, Mr H. Dudeney, an Englishman, to show that M. Lucas had erred, though the two authorities were in agreement concerning the existence of only ten such numbers using the digits in the manner shown, in which the whole number consists in each case of two figures. Here is an example of the latter kind of expression, 82. 3546/197, and the reader, if he choose, may try to find the other nine examples. But the question that forms the main object of this reference is to ask him to find the solitary example of a mixed number using the digits once and once only, in which one figure only appears in place of the 82 shown above, the remaining eight digits forming an improper fraction, the whole in the form of a mixed number being equivalent to 100. A little patience and some ingenuity will enable this to be found, and the more quickly if the trials are methodical.

A CRICKET ’MATCH. Here is a little arithmetical puzzle concerning the scores of a two innings cricket match played in the country where the wickets are not usually the best from the batsmen’s viewpoint, but the non-cricketing reader need not pass it by, for the calculation does not need a would-be solver to have a knowledge of the game either in theory or in practice. X side was sent in to bat first, and the team made the highest total of the four innings which were all played out. The other eleven followed with a score representing one-fifth fewer runs than their opponents in their first effort. But “Z ” showed up bbtter in the second innings for their next visit to the batting creases resulted in not only topping their previous innings score but beating the total made by X in their first attempt by 12 runs. Still that did not enable them to win the match, for X won by 22 runs. Can the reader, without taking up either pen or pencil, find what were the respective scores in the four innings?

DIVIDING £3 4s 2d. There was a balance of £3 4s 2d over after a radio set had been bought with the cheque sent by an uncle of three boys, Archie, Bert, and Cecil, and this sum was, by direction of the donor, divided between the nephew’s in proportion to their ages, which amounted to 35 years. Archie, the eldest of the three, was able to make the required calculation for allotment, and produced the result to his mother, who used it in making the distribution according to her brother’s directions. This is not the way that the calculation was made by the lad, but it will serve to provide an intellect sharpening problem for the reader by finding the exact sum that each of the nephews received. For every eight pence received by Bert, sixpence w’ent to Cecil, and for every shilling that went to the former, one shilling and twopence went to Archie, these unequal proportions being the outcome of the decision to allot the money according to ages of the boys as stated. From these particulars the reader will be able not only to find the sum each lad received but also their respective ages. A CRIBBAGE PUZZLE. Most readers are familiar with the instructive game of cribbage, which, as a mental stimulant, finds favour, with old and young. The game requires at least two players, but here is a little puzzle involving a knowledge of the game, even an elementary one, which can be played alone as in the game called Patience. Place the cards one to nine on the table and arrange them in the form of a. square three by three so that a score of 16 can be counted as the cards lie on the table. It is, of course, clear that there can be no pairs, so that the player must rely upon “ runs ” (sequences) and the adding up of 15, in each of which cases the cards must be' contiguous to one another. For instance if a run of four, five, six is claimed these cards must lie together, the actual direction not being material, and there is no limit to the number of cards forming a sequence under these conditions. The same thing applies to adding up 15, the cards making the 15 must be together. Can the reader arrange the cards in the manner shown so that a score of 16 is possible!

THE AGE OF A TREE, In considering the fallowing question the reader is warned that it contains a little trap. It is suggested that it be read carefully before he attempts to answer it. There are 17 trees in a row on the south boundary of a suburban garden. The first was planted midway on that line many years ago, and at the end of every year after the first planting another one was put in, so that to-day there are eight on each side of the original tree. Had the planting continued in the same way there would now be a much longer row of trees, but the planting was stopped at the seventeenth. At the present time the central tree is exactly four times older than the one last planted, and the interesting question for the reader to answer. from his armchair without bothering with pen or pencil is: What is the present age of the original tree? SOLUTIONS OF LAST WEEK’S PROBLEMS. BLAMING THE MIDDLEMAN. The article in question cost the manufacturer £2. TWO FOR THE ARMCHAIR. Twenty-eight yards, though perhaps some readers made it 30. The answer to the second question is 8 feet 6 inches. THE SHORTEST WAY ROUND. The shortest route measured by time is from the south west corner to a point on the boundary between the two portions of land, exactly 22 yards from the west boundary of the paddock, thence direct" to the north-eastern corner. COLOURED COUNTERS. Twenty-four blue, 2D green, 15 purple, 14 red, and seven yellow. Total 80. SPENT FIFTY PER . CENT. The gentleman started with £l2 in single notes and 16 single shillings, and returned with 12 half notes and four two-shilling pieces. ANSWERS TO CORRESPONDENTS. “ Inquirer.”—The curious reasoning was explained in issue of January 16. A B C. —If the question had not stipulated that the remaining one should occupy its original position at the end, your view would be correct. Thanks. •‘ Colenso.”—One will appear next week.