NUTS TO CRACK

Otago Witness, Issue 3912, 5 March 1929, Page 68

NUTS TO CRACK

By T. L Briton. V'. (Fob in Otago Witnbss.) Readers with a little Ingenuity will find In this column an abundant store ot entertainment and amusement, and the solving of the problems should provide excellent mental exhilaration. While some of the 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 rdutions, unless these are specially asked for, but to keep them for comparison with those published in the issue following the publication of the problems.

A MATHEMATICAL FRUIT DEALER. A certain fruit dealer was known amongst his confreres as a “figure crank,” for whatever transactions he had at the market, they always provided him with material for a little money problem. On one occasion he was the purchaser of four kinds of fruit —plums, pears, apricots, and apples—and it so happened that the whole consignment was sold in bulk at per dozen, the fruit arriving at the market that way owing to a shortage of cases. The dealer decided to buy an equal number of each variety, but on making a calculation at the prices on offer, he found it was possible to buy too much fruit under those conditions. He therefore decided to secure the minimum number that could possibly be obtained by purchasing the same quantity of each kind. Can the reader find by an easy calculation how much the dealer would spend in this way, and the number of each variety, assuming the values were as follows:—Three apples were worth as much as four plums, four pears as much as nine apples, 13 apricots as much as seven pears, seven apples costing three pence?

A CUROSITY. Most readers are aware of many curosities concerning the number 11, and one which came under notice recently is worth recording. In a number consisting of 9 of the ten digits (the cypher in this instance being deemed a digit), if the sum of the individual figures in the odd places is the same as the total of those in the even places, the number in question is divisible by 11 without remainder. For example, 678943012 is a case in point. Another curious feature, not known as well, is that if the difference between the sum of the figures in the odd places, and those in the even places is 11 or a multiple of 11, the same rule applies. With these data to work upon, can the reader find the smallest number containing 9 of the 10 digits (including zero) that can be divided by 11 without remainder ? PERCENTAGE OF PROFIT. Not infrequently questions have been asked by readers concerning the proper method of determining the profit or loss in a trading transaction, and one just received from “Grocer” prompts this explanation and problem. Profit or loss may be reckoned in general terms as a percentage of the money originally spent, and it should be noted that it is calculated on the outlay and not on the amount for which the goods are sold.

Here is a simple little problem which will serve to illustrate this fundamental rule. A storekeeper marks his goods at a price from which he can deduct 74 per cent, and still make a profit of'lo per cent, on his outlay. If an article is marked according to this rule, at £2 4s, can the reader say what is its cost price, remembering that if the article be sold for that amount and cost only half of the sum stated, the percentage of profit is 100, not 50?

AN ACCOMMODATING PEG. Some time ago tlie reader was asked if he could explain how the following njechanical feat was possible:—A brass plate has three apertures, one circular, one square, and one triangular. The experimenter is handed a cork cylindrical in shape which exactly fits the circular hole, and he is required to fashion the cork into such shape that it will exactly, fit any of the three openings in the plate, the diameter of the circle, the height of the triangle, and a side of the square being all equal. “ New Reader ” now asks if the method published some time in 1927 could be given again, and as the number of readers of this column has increased greatly since then, the explanation, which is very useful to know, will hppear next week. In the meantime, Can the reader discover the theory of the method, or demonstrate it practicallyA piece of stiff cardboard will answer the purpose quite as well as brass. BAGS - Of COINS." Ten sealed canvas bags contain ■ coins of various denominations, the value of the contents being different in every case. No bag is of the same size as any other, and curiously, the smaller the bag ,the more valuable are the contents,. all the bags being of graduating, values. .They are numbered in sequence 1

to 10, the latter being the largest and therefore of the lowest value, the bag numbered one being the most costly. If these bags are arranged in two rows in numerical order as follows:—

123 4 5 6 7 8 910 It will be observed that no bag is situated either below or on the right of one of less value than itself. Now, the problem for the reader is to find in how many different ways these bags can be arranged in accordance with the conditions stated? It is an interesting and not difficult calculation in combinations.

LAST WEEK’S SOLUTIONS.

AIR PRESSURE. The cylinder would require to be six inches long for the purpose required. THE CLOCK ON THE MANTELPIECE. The fire would have to be alight for nine hours in the twenty-four. A ROW OF TREES. The first planted tree was 24 years oil at the time stated, and the youngest three years, for seven times older is equivalent to eight times as old. A DIGITAL CROSS. There are 2592 different ways in which the figures can be arranged, exactly eight times as many as the correspondent' stated. A COMBINED JOB. It was necessary to put on 12 extra boys to enable the job to be completed in the extra day stated. ANSWERS TO CORRESPONDENTS. “ Query.”—He need only travel *l9 miles if he started at the point B where the two roads converge. The route can. of course, be varied, but not shortened. “ Mixture.”—The proportion is 40 to 1 of wine and water respectively if your statement is correct, for if the position were reversed it is obvious,• on your own figures, that the ratio would be 1 to 40.

“ Horizon.”—When the sun is in the horizon, it is he length of one-half the earth’s diamet.r farthest distant from any fixed object on the earth that it is at noon when the sun is in its meridian. Thanks for comment.