INTELLECT SHARPENERS.

Otago Daily Times, Issue 20834, 28 September 1929, Page 21

INTELLECT SHARPENERS.

By T. L. Briton. THREE CONSECUTIVE NUMBERS. It was first pointed out by the mathematician Lucius that any three consecutive numbers possess many curious features when submitted to elementary processes of mathematics. It is quite useful to note some of these characteristics. For example, the square of the smallest of any three consecutive numbers added to the middle one gives a total of exactly one more than the product of the first two of those consecutive numbers; also, the square of the middle one added to the other two numbers gives a total of exactly one more than the "first number adddd to the product of the other two. It is a simple matter to verify these theories, but a question bearing upon them which requires a few moments’ thought and calculation is. Supposing the numerical result of one of these axioms in a given case is 359, what are the thre'e consecutive numbers involved? THREE HOUSES SHARE. Thirty gallons 'of home-made wine were placed in half a dozen 10-gallon kegs, to be sent to three houses, A, B, and C, without the contents being interfered with in any way. There were two varieties—ginger wine and cider—and each cask contained a different quantity from the others, but none of them was full. There were no brands on the kegs to indicate which contained the ginger wine and which the apple juice. The only marks were those showing the amount of liquor contained in them. The respective quantities in gallons were:—Three and three-quarters, four and a-quarter, four and three-quarters, five and a-half, five and three-quarters, and sis. The whole 30 gallons were to be so allotted And delivered to the three houses that every occupant would have an equal share. The whole of the ginger wine went to house “ A,” whose inmates numbered exactly two-fifths of the number in house,**" B,” and two-thirds of those in house “C.” Now, as the whole of these ■ conditions were carried out, and a certain number or kegs were delivered intact to the respective bouses, how many persons shared the ginger wine, and which kegs contained the cider? POTATO DIGGING. A simple problem which the reader may elect to solve without the help of pen and pencil concerns the digging of 20 acres of potatoes, but if the mental effort proves, a little strenuous those “ aids ” are not barred. A labourer agreed to dig the field and bag the crop on condition that he received one-third of the number,of filled bags, which were all of uniform weight, the farmer receiving the balance of two-thirds. After the work had been completed, but before the potatoes could be removed, heavy floods occurred, resulting in oneninth of the number of filled bags being wholly destroyed. On the assumption that the labourer agreed to lose his portion of the destroyed crop and receive his share of what was left according to contract, how many bags were originally filled if the labourer took away 384? IN A PLAY AREA. , On a recent afternoon a gentleman drove up" to one’ of the juvenile ,play areas and left with the lady supervisor a basket of cherries to be divided equally amongst the children present. The object of the gentleman’s visit being unknown to the youngsters, the supervisor had the opportunity of counting the cherries unnoticed, and of planning a novel scheme of dividing and distributing them equally, which she was enabled to do with the 870 cherries in the basket. Calling a’ boy up, she handed him sufficient to give every boy and girl one each excepting himself. Youngster No. 2 was then called and similarly instructed; then No. 3, and so on until all the cherries were exhausted and each youngster had made one distribution and each had received an equal number. Here is a little calculation to- be made on these few details. As the distributing child did not receive a cherry when so acting, can the reader say how many children participated in the fruit if the proportion of boys to girls'was as two to one? MORE CHARITY, At the beginning of the winter a person set aside a certain sum of money to btdistributed weekly among necessitous people in the immediate neighbourhood, out the amount was never to be exceeded even though the number of applicants increased. Here is a useful little problem on, the point. A given sum, it may be_ assumed, was distributed last week in this way, each applicant receiving an equal amount. On the assumption that if the number had been two fewer each applicant would automatically have received one shilling more, but if in that week’s dole the number of demands had been three extra, each of the applicants would have received one shilling less, how many applicants shared the charity in that week, and what was the fixed sum given by the generous donor? LAST WEEK’S SOLUTIONS. A DWARFED TREE. At the end of the first year after the tree began to grow again when 12 feet high, its height was 20 feet. A SQUARE SIX BY SIX. The highest number in the, square was 112, and the lowest 7, the respective additions totalling 357. ESTIMATING THE MILEAGE. The distance from Y to Z is 84 miles, Tompkin’s rate of walking being uniformly 21 miles a day. one mile per day more than hjs friend walked on the second day out. The latter’s daily increase was two miles. IMAGINARY AND OTHERWISE. 676. The formula is (25 x 26)2 pins 26, 2 equivalent to the square of the number of letters in the alphabet. A BAKER’S DOZEN AND THE OTHER. Make a six-sided figure with six matches, and with the rest of them six equilateral triangles of equal size can be formed within it, the apexes meeting in the centre of the hexagon. ANSWERS TO CORRESPONDENTS. “Weights.”—(l) A chemist uses apothecaries’ weights, and therefore 12 ounces to the pound is correct. (2) Cannot say if this scale is limited to compounding medicines. “ Not Out.”—The arithmetic of the crecket problem is very simple, and another effort will probably enable you to “ score ” off your own “ bat.” “Hongkong.”—Thanks for item,, which looks like good material for a useful and novel little problem.