NUTS TO CRACK

Otago Witness, Issue 4028, 26 May 1931, Page 20

NUTS TO CRACK

By

T. L. Briton.

(For the Otago Witness.) Readers with a little Ingenuity will .And 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 next 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 WITH A SEVEN SQUARE. Some time ago a correspondent asked whether it is possible in a 36 square to arrange counters of six distinct colours (six of each colour), the counters of each group being numbered one to six, so that the digits in each row, column and two long diagonals will add up the total number of the digits involved, viz., 21, the same colour not appearing twice in any of the directions mentioned. He was informed that it is a mathematical impossibility, yet with a seven square it is not difficult of accomplishment. Can the reader find such an arrangement the digits in this case adding up 28? It will be found that in a seven square, arranged according to these conditions, the same colour or digit will not appear twice in any one diagonal in a “ chess ” sense, —that is to say in any line in which a bishop may be moved.

A STRAIGHT STAIRCASE. A staircase in a seaside boarding house is perfectly straight (without the usual elbow and landing) from the ground to the first floor. There are 15 steps to the first floor exclusive of the one at the top, and a question which offers scope for the exercise of a little ingenuity is as to the number of steps that it is necessary to take in walking, one “rise” at a time, from the ground floor to the landing at the top, with the following stipulations:—A return to the ground floor must be made once after starting; each “tread” of the staircase must be used an equal number of times, the top landing not being deemed a “ tread ” in this connection; and not more than one “ rise ” must be taken at one step. What is the fewest number of steps that need be made to reach the top under these conditions. A “ step ” means moving from the ground floor to one “ tread ” or from one “ rise ” to another, the feat being, deemed to be accomplished upon reaching the top, as it is not required to use that landing an equal number of times with the 15 “ treads.” WITH ARITHMETICAL SIGNS. In a large number of ways, all different, the nine digits may be written with arithmetical signs so that the expression forms an elementary sum, the answer to which is 100, but there are much fewer examples where the digits 1 to 9 are in proper sequence. For instance 12 3 minus 4 minus 5 minus 6 minus 7 plus 8 minus 9 is one such arrangement in which six signs are used, but can the reader find an example where three arithmetical signs only are necessary in forming an expression that will give the result mentioned. There is no bar to the employment of any symbol of arithmetic, but the digits must be in their proper order 1 to 9. If fractions are used the line dividing the numerator from the denominator would be considered a sign. DIVIDING THE RESIDUE. A farmer left to his widow and four sons a square block of land of an area of 2560 acres, the four boundaries running direct north, south, east, and west respectively. A square section comprising one quarter of the area situated in the extreme north-west, upon which the homestead was erected, was bequeathed to the widow, and this section was fenced off from the remainder of the original area, the south-eastern corner post being in the exact centre of the full block. The residue, comprising 1920 acres was left to the four sons to be divided equally between them, but the will contained a stipulation that each should fence his section in exactly the same shape. This was done in symmetrical form in so far that all the subdivision fences were erected parallel to the outside boundaries of the original block, and, as there is only one design of subdivision which will fulfil the conditions stated, can the ' reader in a simple calculation find what length of fencing is involved in dividing the five blocks making separate holdings, and excluding the outside boundaries of • the original area?

QUERIED! A few weeks ago an innocent little armchair problem appeared in this column, and quite a number of letters have since been received from readers questioning the correctness of the solution given. Shorn of its more intricate setting here is the problem. A man tendered a forged bank note for £lOO in the purchase of two articles, one at £6O and the other at £lO. The dealer had to cash the note from the next door hotelkeeper, and the customer received his £3O change, taking with him the more expensive article but leaving the other, which was never called for. The question is, What did the dealer actually

lose if the cost of one of the articles to’ him was £4B and the other £B, the money from tlie publican having been refunded immediately the note was dishonoured? One of the objectors (a fellow journalist by the way) says that he is quite sure the correct answer is £lOB, and not the one given, because the dealer not only lost the cost of the article (£4B) anil the £3O given in change, but he had only £7O left out of £lOO to return to the hotelkeeper, and this meant the loss of another £3O, or £lOB in all. Can the reader put our correspondent on the' right track? SOLUTIONS OF LAST WEEK’S PROBLEMS. A LITTLE LOAN. The borrower received £5 4s, the in-* terest charged being at the rate of 110’ per cent, per annum. A SHUNTING PUZZLE. Twenty-six is the fewest number of “ moves,” and there are several methods l with that number. THE LADIES’ PREROGATIVE. 11,616 ladies, 10,164 spinsters, and? 1452 widows. ONE FOR THE ARMCHAIR. Z’s share falling in through death the 84 acres should be divided between the other two in the proportion laid down in the will., X therefore shouldreceive 48 and Y 36 acres. MARKETING TOGETHER. Kate Adams, Bessie Baker, and Mary Cole. ANSWERS TO CORRESPONDENTS. C. J. W. —In the problem “ In Opposite' Directions/’ the slower ear’s uniform speed was 25 miles an hour. Otherwise the distance is inderterminate. Thanks. L. E. W.—63, 100, 45, and 80 are peculiar differing lengths, but not ungeometrical (2) See above. (3) A “nearly insoluble” one, especially “ sans ” figures, would, make a good “ sharpener ” for the many readers who welcome a challenge to their ingenuity. Key essential for cheeking and timesaving. Thanks.