Intellect Sharpeners.

New Zealand Herald, Volume LXIX, Issue 21136, 19 March 1932, Page 5 (Supplement)

Intellect Sharpeners.

BOY'S POCKET MONEY* 't b? T. t. BRITOS. A boy was given a sum of money by hi® uncle to epend in any way ho desired and* notwithstanding that the amount was somewhat large to be spent in this way, tha boy arrived back with only ono penny* the rest of the gift having been spent afc three different sweet shops v At 'the first shop he spent one penny more than half the sum the uncle gavo him, at the second he got 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 boy's shopping, and the question is that if 1)6 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 chair. INTERESTING NUMBERS. Here is a problem from a correspondent F.S., which the reader, fond of discovering "curiosities in figures, should enjoy. _ if we take the cipher 6 and the-two digits 3 and 7, they can be arranged in three different sets, 037, 370, 703, using each figure in the same position once only, eo that the three sets of numbers thus produced will be in arithmetical progression, which for the benefit of the' non-mathe-matical 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 featuro 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 character-, istic, and it will be interesting to know how long it will take ihe readei; not aware of them, to discover the three other sets. It should not be overlooked that each of the three figures in any number must occupy a,different place in each set, 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 bojhe 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, and this the reader will be able to determine with the details that follow. The lengths of two sides of the glass plate are thirteen inches and fifteen inches respectively, and the base of it fourteen 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 are 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 endsVARIED CONDITIONS. "Phe conditions of a "measure" problem published last month were varied by the unintentional omission of a ciause 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 <loing. The question is again brought forward to invite the reader's consideration to an ingenious idea of the last-men-tioned correspondent, by which it is claimed 1,0 be possible in seven operations. The writer of these notes did not contemplate that the expressed terms of the puzzle could be so strained 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. However, the artifice or strategy necessary to enable the accomplishment of jthe feat in seven operations, as claimed by the correspondent, 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 huaselif whether there"is any method, conforming with the conditions, that will reduce the number of operations from eight to seven. ONE-FIFTE THE SIZE. ' "F.S." has sent a useful 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 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, 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 to afterwards divide the square into five figures of equal area, only one of which, however, must be square. The methoi of arriving at the desired result makfs a pretty as well as an easy calculation. —— - ■ y LAST WEEK'S SOLUTIONS. Shearing Tallies.—99o sheep. Two Problems.—(l). Three gallons. (2). Three half-crowns, four florins and a threepenny piece. Total, 15s 9d. Identifying the Brothers.—B.B., D.G., F.W. and Sam Green, the respective spendings by each pair being 2s, ss, 3s and 6s. Total, 16s. Code With Variation.—The actual letters in the words aro used throughout, and the variation consists of every_ word of one syllable being incorporated in the one which immediately follows. Cricket Averages.—(l). The sixth member spent 2s, which is fivepence less than the average, the others spending 2s 6d 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 <>ettin» T tlio prize with nine for 90, aciuiist " G's 11 for 77. It is not correct-: to take an average of two averages unless there is the same quantity in every set. ANSWERS TO CORRESPONDENTS. C.J.M. —Yes, very suitable—thanks. " Curious." —It is not difficult to arrange the digits one to nine and the cipher 0 in the form of an arithmetical sum in addition the answer to which is one. Decimals aro not required and only (lie plus sign is used. Of course no digi* is repeated in the sum which therefore consists of ten figures. Omitted. —A second question ma- recent problem of an alphabetical sum advertently not answered, to which "ABC" has drawn attention. Ina question is "can the readeriind t.e numerical equivalents of,■ the letters " A.8.C." if the result °. f << BC" by " 8.C." is " A.8.C.. Tha answer is that " A" stands for six and " 8.C." for 25,