INTELLECT SHARPENERS

Otago Daily Times, Issue 21598, 19 March 1932, Page 18

INTELLECT SHARPENERS

Written for the Otago Daily Times By T. L. Briton. ONE-FIFTH THE SIZE. An interesting correspondent, F. S., has sent a useful little problem which may require of the would-be solver a good deal of thought to discover the easy method of finding the solution by arithmetical process, though it would be a simple matter to reach the correct result by using a scale and a drawing compass. Take a sheet of paper, and make a diagram of a square of indefinite size, and cut from it, or better still draw within it another square of exactly one-fifth the size of the original figure. There is a condition attached to the question which may render it not quite so easy as it looks—namely, that no part of the smaller square must touch any part of the other. This, of course, precludes using any part of the boundaries of the larger figure as an external line or lines of the required square. There is, of course, more than one method of doing this, but the reader will no doubt choose the most symmetrical form so that his calculation will enable him to divide the square afterwards into five figures of equal area, only one of which, however, must be square. The method of arriving at the desired result makes a pretty as well as an easy calculation. INTERESTING NUMBERS. Here is another one, of a different variety, from the same source which the reader, fond of discovering curiosities in figures, should enjoy. If we take the cipher 0 and the two digits 3 and 7, they can be arranged in three different sets —037, 370, 703 —each figure being used in the same position once only, so that the three sets of numbers thus produced will be in arithmetical progression, which for the benefit of the nonmathematical reader, means that the difference between the first two sets of numbers is the same as that between the second and third, namely, 333. This curious feature is to be found in three other sets of three figures, and notwithstanding that in two instances entirely different digits form the numbers, and in the third example some are different, the common difference is exactly the same, namely, 333. There are no other instances of a set of three digits in a number having this queer and rather uncanny characteristic, and it will be interesting to know how long it will take the reader who is not aware of them, to discover the three other sets. It should hot be overlooked that each of the three figures in any number must occupy a different place in each set.

A BOY’S POCKET MONEY. A boy was given a certain sum of money by his uncle to spend in any way he desired, and, notwithstanding that the amount was somewhat large to be spent in this way, the boy arrived back with only one penny, the rest of the gift having been spent at three different sweets shops. At the first he spent one penny more than half the sum the uncle gave him, at the second he managed to get rid of twopence more than one-half the sum that he had left, and at the third he expended In the same way threepence more than half the sum remaining after leaving the second shop. That concluded the boys’ shopping, and the question is, If he then had only one penny left, how much did the uncle give him, that being the only money the lad had? The reader will of course treat this question as one for the armchair. A TRIANGULAR PLATE, I have a triangular plate of ground glass which evidently has been in use for some previous purpose, for it has a small, round hole bored through it in each of the three corners. With a certain object in view I desire to drill a similar hole in the exact centre, or, rather in such a position that the distance from it to each of the three corners of the plate will be the same, and to pass a flexible steel wire, making it taut, from one corner direct to the central hole through which it passes and thence back to the point of starting, the question is: What length of wire will be required? This the reader will be able to determine with the details that follow; —The lengths of two sides of the glass plate are 13 inches and 15 inches respectively, and the base of it 14 inches. To make an easier calculation of the problem, which, by the way will be; found a very useful one, it may be assumed that the three holes now in the plate arc at the extreme corners, that the plate has no perceptible thickness, and that the length of the required wire will be merely the distance from the proposed hole in the centre to a corner and back, ignoring any length of wire that may be necessary for tying or joining the ends. VARIED CONDITIONS. The conditions of a “ measure ” problem published last month were varied by an unintentional omission of a clause stipulating alternate pourings from the three vessels, whether on the ground or into the other vessels. Under the conditions as published the feat can be accomplished in as few as eight pourings, which Kaera,” “W. E. H.”, and “ L. D.,” three correspondents, succeeded in doing. The question is again brought forward to invite the readers’ consideration to an ingenious idea of “ L. D„” by which it is claimed that the feat is possible in seven operations. The writer of these notes did not contemplate that the expressed terms of the puzzle could be so strained as to permit of the view of “ L. D.,” and thinks that if the conditions already stated are capable of being interpreted to permit of the certain method suggested, the spirit of the question as submitted does not. The artifice or strategy, however, necessary to enable the accomplishment of the feat in seven operations, as claimed by the correspondent mentioned, reveals an original and ingenious idea, and for that reason the question is again referred to the reader to examine the terms of the problem in this new light and decide for himself whether there is any method (conforming with the conditions) that will reduce the number of operations from eight to seven. SOLUTIONS OF LAST WEEK’S PROBLEMS. A CODE WITH VARIATION. The actual letters in the words are used throughout, and the variation consists of every word of one syllable being incorporated in the one which immediately follows. IDENTIFYING THE BROTHERS. 8.8., D.G., F.W., and Sam Green, the respective spendings by each pair being 2s, ss, 3s, and Gs; total, 10s. • AVERAGES. (1) The sixth member spent 2s, which is 5d less than the average, the others spending 2s Od each. (2) “G” secures the trophy with an average of seven. It seems paradoxical, but to take the average of the two averages would result in “ J ” getting the prize with nine for ninety against “ G’s ” eleven for seventy-seven. It is not correct to take an average of two averages unless there is the same quantity in every set. ALSO SOLUBLE MENTALLY. (1) Three gallons. (2) Three halfcrowns, four florins and a threepenny piece; total, 15s 9d.

SHEARING TALLIES, ; 990 sheep. ANSWERS TO CORRESPONDENTS. “ Curious.”—lt is not difficult to arrange the digits 1 to 9 and the cipher o in the form of an arithmetical sum in addition the answer to which is 1. Decimals are not required and only the plus sign is used. Of course no digit is repeated in the sum, which therefore consists of ten figures. C. J. W. —Yes, very suitable. Thanks. A. B. C. has drawn attention to the fact that a second question in a recent problem of an alphabetical sum was inadvertently not answered. The question is, “Can the reader find the numerical equivalents of the letters a. b. c. if the result of multiplying be by be is abc: The answer is that “ a ” stands for six and “ be " for 25.