Part 11

“STEWING.”

38. A student reads two lines more of “Virgil” each day than he did the day before, and finds that, having read a certain quantity in 18 days, he will read at this rate the same quantity in the next 14 days. How much will he read in the whole time?

39. Two bootmakers who lived in the town of B., thrown out of employment, resolved to go to G., a town 24 miles north from B., where there is a large factory; one of them went straight on to G., but the other went first to C., a small township west of B., and then went direct to G., his whole journey being 45 miles. What is the distance from C. to G.?

40. A tree which grows each year 1 inch less than the previous year, grew a yard in the first year; the value of the tree at any time is equal to the number of pence in the cube of the number of yards of its height. What is the value of the tree when done growing?

THIS OFTEN “STICKS” PEOPLE UP.

41. What two odd numbers multiplied together make 7?

MAGIC SQUARES.

A Magic Square is a series of figures arranged in the equal divisions of a square in such a manner that the figures in each row when added up, whether horizontally, vertically, or diagonally, form exactly the same sum.

They have been called “Magic” because the ancients ascribed to them great virtues, and because this arrangement of numbers formed the basis and principle of their talismans. Archimedes devoted a great amount of attention to them, which has caused a great many to speak of them as “the squares of Archimedes.” They may be either odd or even. When the former, the following method will be found valuable:--

With the digits from 1 to 25 form a square so that the numbers when added up horizontally, vertically, or diagonally will amount to 65.

Method.--Imagine an exterior line of squares above the magic square you wish to form, and another on the right hand of it. These two imaginary lines are shown in the diagram.

18 25 2 9 +----+----+----+----+----+ | 17 | 24 | 1 | 8 | 15 | 17 +----+----+----+----+----+ | 23 | 5 | 7 | 14 | 16 | 23 +----+----+----+----+----+ | 4 | 6 | 13 | 20 | 22 | 4 +----+----+----+----+----+ | 10 | 12 | 19 | 21 | 3 | 10 +----+----+----+----+----+ | 11 | 18 | 25 | 2 | 9 | +----+----+----+----+----+

1st. In placing the numbers in the square, we must go in the ascending diagonal direction from left to right, any number which, by pursuing this direction, would fall into the exterior line must be carried along that line of squares, whether vertical or horizontal, to the last square. Thus, 1 having been placed in the centre of the top row, 2 would fall into the exterior square above the fourth vertical line; then ascending diagonally 3 falls into the square diagonally from 2, but 4 falls out of it to the end of a horizontal line, and it must be carried along that line to the extreme left and there placed. Resuming our diagonal ascension to the right we place 5 where the reader sees it, and would place 6 in the middle of the top row, but as we find 1 is already there we look for the direction to

2nd. That when in ascending diagonally we come to a square already occupied, we must place the number which, according to the 1st rule should go into that occupied square directly under the last number placed: thus, in ascending with 4, 5, 6, the 6 must be placed under the 5, because the square next to 5 in diagonal direction is occupied.

A Promising Sign--I O U.

HOW TO FIND THE TOTAL OF A ROW OF FIGURES IN A MAGIC SQUARE.

Rule.--Multiply half the sum of the extremes by the square root of the greatest extreme.

Referring to the example given above, we see that the extremes 1 and 25 added equal 26--half of which is 13; this multiplied by 5 (the square root of 25) gives 65 as the total for each row.

Again, in the next question, the two extremes 1 and 81 equal 82, half of this sum is 41, which multiplied by 9 (the square root of 81) gives 369 as the total for each row.

42. Arrange the figures from 1 to 81 in a square that when added up horizontally, vertically, or diagonally the sum will be 369.

HOW THEY WORKED IT.

Mick and Pat, working in the country some distance from a hotel, arranged with the landlord to take to their hut a small keg of rum. They were unable to pay for the liquor at the time, having only one threepenny piece between them; but Mick proposed that every time he had a drink he would give Pat threepence, and Pat also agreed to pay Mick for his drinks, the cash thus gathered to be brought to the publican when the keg was empty. This proposal was accepted by the publican, the keg of rum handed over to the two Irishmen, who immediately started on their journey. They had not proceeded very far before their burden made them thirsty. Mick is the first to pull up with: “Hold on, Pat, I think I’ll have a drink.” “Begorra,” replied Pat, “you’ll have to pay me for it then.” Mick hands the 3d. to Pat before having a good “pull.” Pat now being the possessor of the price of a drink, slakes his thirst by paying Mick 3d. for it. This form of payment is kept up till the rum has disappeared. On their next visit to the hotel, the 3d piece is handed to the landlord as being payment, according to terms of agreement adopted by him.

43. Arrange the figure’s from 1 to 9 in a square, so that they will add up to 15, horizontally, vertically, or diagonally.

44.

45. A man sold a horse for £35 and half as much as he gave for it, and gained thereby 10 guineas. What did he pay for the horse?

THE DISHONEST SERVANT.

46. A gentleman having bought 28 bottles of wine, and suspecting his servant of tampering with the contents of the wine cellar, caused these bottles to be arranged in a bin in such a way as to count 9 bottles on each side. Nothwithstanding this precaution, the servant in two successive visits stole 8 bottles--4 each time--re-arranging the bottles each time so that they still counted 9 on a side. How did he do it?

+-------------+ | 2 5 2 | | | | | | 5 5 | | | | | | 2 5 2 | +-------------+