PART II. (continued 3)

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27

is solved by CLIO, one of the nine Muses.

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28.

The man who paid a penny on Monday morning to cross the ferry, spent half of what money he then had left in the town, and paid another penny to recross the ferry, and who repeated this course on each succeeding day, reaching home on Saturday evening with one penny in his pocket, started on Monday with £1 1s. 1d. in hand.

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29.

When the three men agreed to share their mangoes equally after giving one to the monkey, and when each helped himself to a third after giving one to the monkey, without knowing that anyone had been before him, and they finally met together, gave one to the monkey, and divided what still remained, there must have been at least seventy-nine mangoes for division at the first.

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30.

If, after having looked at my watch between 4 and 5, I look again between 7 and 8, and find that the hour and minute-hands have then exactly changed places, it was 36¹²⁄₁₃ minutes past 4 when I first looked. At that time the hour-hand would be pointing to 23¹⁄₁₃ minutes on the dial, and at 23¹⁄₁₃ minutes past 7 the hour hand would be pointing to 36¹²⁄₁₃ minutes.

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31.

The number consisting of 22 figures, of which the last is 7, which is increased exactly sevenfold if this 7 is moved to the first place, is 1,014,492,753,623,188,405,797.

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32.

The two sacks of wheat, each 4 feet long and 3 feet in circumference, which the farmer sent to the miller in repayment for one sack 4 feet long and 6 feet in circumference, far from being a satisfactory equivalent, contained but half the quantity of the larger sack, for the area of a circle the diameter of which is double that of another is equal to four times the area of that other.

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33.

The five gamblers, who made the condition that each on losing should pay to the others as much as they then had in hand, and who each lost in turn, and had each £32 in hand at the finish, started with £81, £41, £21, £11, and £6 respectively.

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34.

If we know the square of any number, we can rapidly determine the square of the next number, without multiplication, by adding the two numbers to the known square. Thus if we know that the square of 87 is 7569,

then the square of 88 = 7569 + 87 + 88 = 7744; so too the square of 89 = 7744 + 88 + 89 = 7921; and the square of 90 = 7921 + 89 + 90 = 8100.

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35.

The two numbers which solve the problem—

are 1³⁄₈ and 9⁵⁄₈, for 1³⁄₈ + 9⁵⁄₈ = 11, and ⁷⁷⁄₈ ÷ ¹¹⁄₈ = 7.

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36.

There must be nine things of each sort, in order that 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 different selections may be made from twenty sorts of things.

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37.

The women who had respectively 33, 29, and 27 apples, and sold the same number for a penny, receiving an equal amount of money, began by selling at the rate of three a penny. The first sold ten pennyworth, the second eight pennyworth, and the third seven pennyworth.

The first had then left three apples, the second five, and the third six. These they sold at one penny each, so that they received on the whole—

The first 10d. + 3d. = 13d.

The second 8d. + 5d. = 13d.

The third 7d. + 6d. = 13d.

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38.

The puzzle—

is solved by fie, seen.

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39.

If a bun and a half cost three halfpence, it is plain that each bun costs a penny, but, by general custom, you buy seven for sixpence.

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40.

The hands of a watch would meet each other twenty-five times in a day, if the minute-hand moved backwards and the hour-hand forwards. They are, of course, together at starting.

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41.

The only way in which half-a-crown can be equally divided between two fathers and two sons, so that a penny is the smallest coin made use of, is to give tenpence each to a grandfather, his son, and his grandson.

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42.

If the number of the revolutions of a bicycle wheel in six seconds is equal to the number of miles an hour at which it is running, the circumference of the wheel is 8⁴⁄₅ feet.

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43.

The hour that struck was twelve o’clock.

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44.

Sixty years.

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45.

If I jump off a table with a 20lb dumb-bell in my hand there is no pressure upon me from its weight while I am in the air.

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46.

If at a bazaar I paid a shilling on entering each of four tents, and another shilling on leaving it, and spent in each tent half of what was in my pocket, and if my fourth payment on leaving took my last shilling, I started with 45s., spending 22s. in tent 1, 10s. in tent 2, 4s. in tent 3, and 1s. in tent 4, having also paid to the doorkeepers 8s.

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47.

When rain is falling vertically at 5 miles an hour, and I am walking through it at 4 miles an hour, the rain drops will strike the top of my umbrella at right angles if I hold it at an angle of nearly 39 degrees.

As I walk along, meeting the rain, the effect is the same as it would be if I was standing still, and the wind was blowing the rain towards me at the rate of 4 miles an hour.

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48.

When one monkey descends from the top of a tree 100 cubits high, and makes its way to a well 200 yards distant, while another monkey, leaping upwards from the top, descends by the hypotenuse to the well, both passing over an equal space, the second monkey springs 50 cubits into the air.

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49.

The steamboat which springs a leak 105 miles east of Tynemouth Lighthouse, and, putting back, goes at the rate of 10 miles an hour the first hour, but loses ground to the extent in each succeeding hour of one-tenth of her speed in the previous hour, never reaches the lighthouse, but goes down 5 miles short of it.

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50.

Twenty-one hens will lay ninety-eight eggs in a week, if a hen and a-half lays an egg and a-half in a day and a-half. Evidently one egg is laid in a day by a hen and a-half, that is to say three hens lay two eggs in a day. Therefore, twenty-one hens lay fourteen eggs, in a day, or ninety-eight in a week.

Q. E. D. (Quite easily done!)

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51.

If the population of Bristol exceeds by 237 the number of hairs on the head of anyone of its inhabitants that are not bald, at least 474 of them must have the same number of hairs on their heads.

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52.

In tipping his nephew from seven different coins, the uncle may give or retain each, thus disposing of it in two ways, or of all in 2 × 2 × 2 × 2 × 2 × 2 × 2 ways. But as one of these ways would be to retain them all, there are not 128, but only 127 possible variations of the tip.

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53.

The prime number which fulfils the various conditions of the question is 127. Increased by one-third, excluding fractions, it becomes 169, the square of 13. If its first two figures are transposed, and it is increased by one-third, it becomes 289, the square of 17. If its first figure is put last, and it is increased by one-third, it becomes 361, the square of 19. If, finally, its three figures are transposed, and then increased by one-third, it becomes 961, the square of 31.

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54.

Six things can be divided between two boys in 62 ways. They could be carried by two boys in 64 ways (2 × 2 × 2 × 2 × 2 × 2), but they are not divided between two boys if all are given to one, so that two of the 64 ways must be rejected.

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55.

The highest possible score that the dealer can make at six cribbage, if he is allowed to select the cards, and to determine the order of play, is 78. The dealer and his opponent must each hold 3, 3, 4, 4, the turn-up must be a 5, and crib must have the knave of the suit turned up, and 5, 5, 5. It will amuse many of our readers to test this with the cards.

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56.

The picture frame must be 3 inches in width all round, if it is exactly to equal in area the picture it contains, which measures 18 inches by 12 inches.

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57.

If my mother was 20 when I was born, my sister is two years my junior, and my brother is four years younger still, our ages are 56, 36, 34, and 30.

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58.

The spider in the dockyard, whose thread was drawn from her by a revolving capstan 1 foot in diameter, until 73 feet of it were paid out, after walking for a mile round and round the capstan at the end of the stretched thread in an effort to unwind it all, had, when she stopped in her spiral course, 49 more feet to walk to complete her task.

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59.

The mountebank at a fair, who offered to return any stake a hundredfold to anyone who could turn up all the sequence in twenty throws of dice marked each on one face only with 1, 2, 3, 4, 5, or 6, should in fairness have engaged to return 2332 times the money; for of the 46,656 possible combinations of the faces of the dice, only one can give the six marked faces uppermost. Thus the chance of throwing them all at one throw is expressed by ¹⁄₄₆₆₅₆, and in twenty throws by about ¹⁄₂₃₃₂.

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60.

If 90 groats (each = 4d.) feed twenty cats for three weeks, and five cats consume as much as three dogs, seventy-two hounds can be fed for £39 in a period of ninety-one days.

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61.

When equal wine-glasses, a half and a third full of wine, are filled up with water, and their contents are mixed, and one wine-glass is filled with the mixture, it contains ⁵⁄₁₂ wine and ⁷⁄₁₂ water.

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62.

The arrangement by which St Peter is said to have secured safety for the fifteen Christians, when half of the vessel’s passengers were thrown overboard in a storm, is as follows:—

XXXXIIIIIXXIXXXIXIIXXIIIXIIXXI

Each Christian is represented by an X, and if every ninth man is taken until fifteen have been selected, no X becomes a victim.

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63.

If Farmer Southdown’s cow had a fine calf every year, and each of these, and their calves in their turn, at two years old followed this example, the result would be no less than 2584 head in sixteen years.

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64.

The number of the flock was 301. This is found by first taking the least common multiple of 2, 3, 4, 5, 6, which is 60, and then finding the lowest multiple of this, which with 1 added is divisible by 7. This 301 is exactly divisible by 7, but by the smaller numbers there is 1 as remainder.

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65.

The rule for determining easily the number of round bullets in a flat pyramid, with a base line of any length, is this:—

Add a half to half the number on the base line, and multiply the result by the number on that line. Thus, if there are twelve bullets as a foundation—

12 + ¹⁄₂ = ¹³⁄₂; and ¹³⁄₂ × ¹²⁄₁ = 78.

The same result is reached by multiplying the number on the base line by a number larger by one, and then halving the result. Thus—

12 × 13 = 156, 156 ÷ 2 = 78.

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66.

We can gather from the lines—

that old General Host had an army 24,000 strong.

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67.

When the farmer sent five pieces of chain of 3 links each, to be made into one continuous length, agreeing to pay a penny for each link cut, and a penny for each link joined, the blacksmith, if he worked in the best interest of the farmer, could only charge sixpence: for he could cut asunder one set of 3 links, and use these three single links between the other four sets.

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68.

If, in a parcel of old silver and copper coins, each silver piece is worth as many pence as there are copper coins, and each copper coin is worth as many pence as there are silver coins, there are eighteen silver and six copper coins, when the whole parcel is worth eighteen shillings.

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69.

These are five groups that can be arranged with the numbers 1 to 11 inclusive, so that they are all equal:—

(8² - 5² + 1) = (11² - 9²) = (7² - 3²) = (6² + 2²) = 4(10).

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70.

John Bull, under the conditions given, lived to the age of eighty-four years.

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71.

The two numbers to each of which, or to the halves of which, unity is added, forming in every case a square number, are 48 and 1680.

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72.

The true weight of a cheese that seemed to weigh 16 ℔s. in one scale of a balance with arms of unequal length, and only 9℔s. in the other, is 12℔. This is found by multiplying the 16 by the 9, and finding the square root of the result.

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73.

The two parts into which 100 can be divided, so that if one of them is divided by the other the quotient is again exactly 100 are 99¹⁄₁₀₁ and ¹⁰⁰⁄₁₀₁.

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74.

If, with marbles in two pockets, I add one to those in that on the right, and then multiply its contents by the number it held at first, and after dealing in a similar way with those on the left, find the difference between the two results to be 90; while if I multiply the sum of the two original quantities by the square of their difference the result is 176, I started with twenty-three in the right-hand pocket and twenty-one in the other.

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75.

The circle of twenty-one friends who arranged to meet each week five at a time for Bridge so long as exactly the same party did not meet more than once, and who wished to hire a central room for this purpose, would need it for no less than 20,349 weeks, or more than 390 years, to carry out their plan.

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76.

If a herring and a half costs (not cost) a penny and a half, the price of a dozen such quantities is eighteenpence.

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77.

The sum of money which in a sense appears to be the double of itself is 1s. 10d., for we may write it one and ten pence or two and twenty pence.

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78.

The “comic arithmetic” question set by Dr Bulbous Roots—

Divide my fifth by my first, and you have my fourth; subtract my first from my fifth, and you have my second; multiply my first by my fourth followed by my second, and you have my third; place my second after my first, and you have my third multiplied by my fourth—is solved by COMIC.

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79.

If the earth could stand still, and a straight tunnel could be bored through it, a cannon ball dropped into it, if there is no air or other source of friction, would oscillate continually from end to end.

Taking air into account, the ball would fall short of the opposite end at its first lap, and in succeeding laps its path would become shorter and shorter, until its initial energy was exhausted, when it would come to rest at the centre.

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80.

He sent 163. She sent 157.

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81.

When twins were born the estate was properly divided thus:—

Taking the daughter’s share as 1

The widow’s share would be 2

And the son’s share 4

Total 7 shares.

So the son takes four-sevenths, the widow two-sevenths, and the daughter one-seventh of the estate.

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82.

If each of my strides forwards or backwards across a 22 feet carpet is 2 feet, and I make a stride every second; and if I take three strides forwards and two backwards until I cross the carpet, I reach the end of it in forty-three seconds. In three steps I advance 6 feet. Then in two steps I retrace 4 feet, thus gaining only 2 feet in five steps, i.e., in five seconds. I therefore advance 16 feet in forty seconds, and three more strides cover the remaining 6 feet.

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83.

If the captain of a vessel chartered to sail from Lisbon to New York, which appear on a map of the world to be on the same parallel of latitude, and which are, along the parallel, about 3600 miles apart, takes his ship along this parallel, he will not be doing his best for the impatient merchant who has had an urgent business call to New York.

The shortest course between the two points is traced by a segment of a “great circle,” having its centre at the centre of the earth, and touching the two points. This segment lies wholly north of the parallel, and is the shortest possible course.

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84.

When John and Harry, starting from the right angle of a triangular field, run along its sides, and meet first in the middle of the opposite side, and again 32 yards from their starting point, if John’s speed is to Harry’s as 13 to 11, the sides of the field measure 384 yards.

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85.

If two sorts of wine when mixed in a flagon in equal parts cost 15d., but when mixed so that there are two parts of A to three of B cost 14d., a flagon of A would cost 20d., and a flagon of B 10d.

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86.

If, when a man met a beggar, he gave him half of his loose cash and a shilling, and meeting another gave him half what was left and two shillings, and to a third half the remainder and three shillings, he had two guineas at first.

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87.

The clerk who has two offers of work from January 1, one from A of £100 a year, with an annual rise of £20, and the other from B of £100 a year, with a half-yearly rise of £5, should accept B’s offer.

The half-yearly payments from A (allowing for the rise), would be 50, 50, 60, 60, 70, 70, etc., etc.; and from B they would be 50, 55, 60, 65, 70, 75, etc., etc., so that B’s offer is worth £5 a year more than A’s always.

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88.

If I have a number of florins and half-crowns, but no other coins, I can pay my tailor £11, 10s. in 224 different ways.

This can be found thus by rule of thumb: Start with 0 half-crowns and 115 florins. Then 4 half-crowns and 110 florins. Add 4 half-crowns and deduct 5 florins each time till 92 half-crowns and 0 florins is reached.

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89.

The monkey climbing a greased pole, 60 feet high, who ascended 3 feet, and slipped back 2 feet in alternate seconds, reached the top in 1 minute, 55 seconds, for he did not slip back from the top.

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90.

When Adze, the carpenter, secured his tool-chest with a puzzle lock of six revolving rings, each engraved with twelve different letters, the chances against any one discovering the secret word formed by a letter on each ring was 2,985,983 to 1; for the seventy-two letters may be placed in 2,985,984 different arrangements, only one of which is the key.

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91.

The five married couples who arranged to dine together in Switzerland at a round table, with the ladies always in the same places, so long as the men could seat themselves each between two ladies, but never next to his own wife, were able under these conditions to enjoy thirteen of these nights at the round table.

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92.

If in a calm the tip of a rush is 9 inches above the surface of a lake, and as the wind rises it is gradually blown aslant, until at the distance of a yard it is submerged, it is growing in water that is 5 feet 7¹⁄₂ inches deep.

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93.

Aminta was eighteen.

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94.

When Dick took a quarter of the bag of nuts, and gave the one over to the parrot, and Tom and Jack and Harry dealt in the same way with the remainders in their turns, each finding a nut over from the reduced shares for the bird, and one was again over when they divided the final remainder equally, there were, at the lowest estimate, 1021 nuts in the bag.

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95.

Eight and a quarter is the answer to the nonsense question—

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96.

The similar fraction of a pound, a shilling, and a penny which make up exactly a pound are as follows:—

s. d.

²⁴⁰⁄₂₅₃ of £1 = 18 11 ¹⁶⁹⁄₂₅₃

²⁴⁰⁄₂₅₃ of 1s. = 11 ⁹⁷⁄₂₅₃

²⁴⁰⁄₂₅₃ of 1d. = ²⁴⁰⁄₂₅₃

£1 0 0

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97.

When Dr Tripos thought of a number, added 3, divided by 2, added 8, multiplied by 2, subtracted 2, and thus arrived at double the number, he started with 17.

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98.

When A and B deposited equal stakes with C, and agreed that the one who should first win three games of billiards should take all, but consented to a division in proper shares when A had won two games and B one, it was evident that if A won the next game all would go to him, while if he lost he would be entitled to one half. One case was as probable as the other, therefore he was entitled to half of these sums taken together; that is, to three quarters of the stakes, and B to a quarter only.

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99.

The average speed of a motor which runs over any course at 10 miles an hour, and returns over the same course at 15 miles an hour, is 12 miles an hour, and not 12¹⁄₂, as might be imagined. Thus a run of 60 miles out takes, under the conditions, six hours, and the return takes four hours; so that the double journey of 120 miles is done in ten hours, at an average speed of 12 miles an hour.

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100.

Farmer Hodge, who proposed to divide his sheep into two unequal parts, so that the larger part added to the square of the smaller part should equal the smaller part added to the square of the larger part, had but one sheep.

Faithful to his word, he divided this sheep into two unequal parts, ²⁄₃ and ¹⁄₃, and was able to show that ²⁄₃ + ¹⁄₉ = ⁷⁄₉, and that ¹⁄₃ + ⁴⁄₉ = ⁷⁄₉. He was heard to declare further, and he was absolutely right, that no number larger than 1 can be so divided as to satisfy the conditions which he had laid down.

The fact that sheep is both singular and plural, adds much to the perplexing points of this attractive problem.

Here is a very simple proof that the number must be 1:—

Let a + b = no. of sheep

then a² + b = b² + a

a² - b² = a - b

or (a + b)(a - b) = a - b

therefore a + b = 1.

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101.

A horse that carries a load can draw a greater weight up the shaft of a mine than a horse that bears no burden. The load holds him more firmly to the ground, and thus gives him greater power over the weight he is raising from below.

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102.

In the six chests, of which two contained pence, two shillings, and two pounds, there must have been at least the value of 506 pence. This can be divided into 22 (or 19 + 3) shares of 23d. each, or 23 (19 + 4) shares of 22d. each. Evidently then the treasure can be divided so that 19 men have equal shares, while their captain has either 3 shares or 4 shares.

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103.

If I bought a parcel of nuts at 49 for 2d., and divided it into two equal parts, one of which I sold at 24, the other at 25 a penny; and if I spent and received an integral number of pence, but bought the least possible number of nuts, I bought 58,800 nuts, at a cost of £10, and I gained a penny.

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104.

When, with a purse containing sovereigns and shillings, after spending half of its contents, I found as many pounds left as I had shillings at first, I started with £13, 6s.

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105.

When the lady replied to a question as to her age—

she was 28 years old.

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106.

If cars run, at uniform speed, from Shepherd’s Bush to the Bank, at intervals of two minutes, and I am travelling at the same rate in the opposite direction, I shall meet 30 in half-an-hour, for there are already 15 on the track approaching me, and 15 are started from the other end during my half hour’s course.

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107.

If it was possible to carry out my offer of a farthing for every different group of apples which my greengrocer could select from a basket of 100 apples, he would be entitled to the stupendous sum of £18,031,572,350 19s. 2d.

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108.

If the minute-hand of a clock moves round between 3 and 4 in the opposite direction to the hour-hand, the hands will be exactly together when it is really 41⁷⁄₁₃ minutes past 3.

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109.

If the walnut monkey had stopped to help the other, and they had eaten filberts at equal rates, they would have escaped in 2¹⁄₄ minutes.

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110.

The value of the cheque, for which the cashier paid by mistake pounds for shillings, was £5, 11s. 6d. The receiver to whom £11, 5s. 6d. was handed, spent half-a-crown, and then found that he had left £11, 3s., just twice the amount of the original cheque.

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111.

The number 14 can be made up by adding together five uneven figures thus:—11 + 1 + 1 + 1. It will be seen that although only four numbers are used, 11 is made up of two figures.

Here is another, and quite a curious solution, 1 + 1 + 1 + 1 = 4, and with another 1 we can make up 14!

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112.

A business manager can fill up three vacant posts of varying value from seven applicants in 210 different ways. For the first post there would be a choice among 7, for the second among 6, and for the third among 5, so that the possible variations would amount to 7 × 6 × 5 = 210.

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113.

If the fasting man, who began his task at noon, said it is now ⁵⁄₁₁ of the time to midnight, he spoke at 3.45 p.m., meaning that ⁵⁄₁₁ of the remaining time till midnight had elapsed since noon.

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114.

If a clock takes six seconds to strike 6, it will take 12 seconds to strike 11, for there must be ten intervals of 1¹⁄₅ seconds each.

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115.

Twenty horses can be arranged in three stalls, so that there is an odd number in each, by placing one in the first stall, three in the second, and sixteen (an odd number to put into any stall!) in the third.

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116.

The little problem, “Given a, b, c, to find q,” is solved, without recourse to algebra, thus: a, b, c, = c, a, b; take a cab and go over Kew Bridge, and you find a phonetic Q!

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117.

Tom Evergreen was 75 years old when he was asked his age by some men at his club in 1875, and said—“The number of months that I have lived are exactly half as many as the number which denotes the year in which I was born.”

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118.

Eight different circles can be drawn. A circle can have one of the three inside and two outside in three ways, or one outside and three inside in three ways (each of the three being inside or outside in turn), or all three may be inside, or all three may be outside, the touching circle.

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119.

The way to arrange 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, so that used once each they form a sum which is equal to 1 is this:—

3570 + 148296 = 1

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120.

The sum of the first fifty numbers may be found without any addition thus:—The first fifty numbers form twenty-five pairs of fifty-one each (1 + 50, 2 + 49, etc., etc.), and 51 × 25 is practically 51 × 100 ÷ 4 = 1275.

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121.

The tramcar A, which started at the same time as B, but ran into a “lie by” in four minutes, and waited there five minutes till B came along, when they completed their courses at the same moment in opposite directions, could have run the whole distance in ten minutes.

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122.

What remains will be 8 if we take 10 and double it by writing one 10 over another so as to form 18, and then deduct 10.

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123.

If the average weight of the Oxford crew is increased by 2℔s., when one of them who weighs 12 stone, is replaced by a fresh man, the weight of that substitute is 13 stone 2℔s.

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124.

If a motor-car is twice as old as its tyres were when it was old as its tyres are, and if, when these tyres are as old as the car itself is now, their united ages will be 2¹⁄₄ years, the car is now 12 months old, and the tyres have had 9 months’ wear.

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125.

A and B, who could each carry provisions for himself for twelve days, started to penetrate as far as possible into a desert, on the understanding that neither of them should miss a day’s food. After an advance of four days, each had provisions still for eight days. One gave four portions of his store to his companion, which did not overload him, and returned with the other four. His comrade was then able to advance another four days’ journey, and still have rations for the eight days’ return. Thus the furthest possible penetration into the desert under the conditions was an eight days’ march.

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126.

If, when a bottle of medicine and its cork cost half-a-crown, the bottle and the medicine cost two and a penny more than the cork, the cork cost twopence half-penny.

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127.

A boat’s crew far from land, with no sail or oars, and with no assistance from wind or stream, or outside help of any kind, can regain the shore by means of a coil of rope. Motion is given to the boat by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. This curious illustration of mechanical principles is from “Ball’s Mechanical Recreations.” (Macmillan.)

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128.

It will be found that after a crown and as many four-shilling pieces as possible have been crammed into our pockets, there would still be room for one sixpence and one threepenny-piece in some corner or cranny. We can, therefore, have one crown, one sixpence, one threepenny-piece, and as many four-shilling pieces as our pockets will hold, and yet be unable to give change for a half-sovereign.

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129.

There were fifteen apples in the basket. Half of these and half an apple, i.e., eight were first given, then half the remainder and half an apple, i.e., four, then on similar lines two, leaving one in the basket.

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130.

The Queer Division—

is solved by LV ÷ V, or 55 ÷ 5 = 11.

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131.

A motor that goes 9 miles an hour uphill, 18 miles an hour downhill, and 12 miles an hour on the level, will take 8 hours and 20 minutes to run 50 miles out and return at once over the same course.

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132.

The number of shots fired at a mark was 420 each by A, B, and C. A made 280 hits, B 315, and C 336.

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133.

If a dog and a cat, evenly matched in speed, run a race out and back over a course of 75 yards in all, and the dog always takes 5 feet at a bound, and the cat 3 feet, the cat will win, because at the turning point the dog overleaps the half distance more than the cat does, and so has a longer run in.

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134.

When a man caught up a wagon going at 3 miles an hour, which was just visible to him in a fog at a distance of 55 yards, and which he saw for five minutes before reaching it, he was walking at the rate of 3³⁄₈ miles an hour.

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135.

Three horses, A B C, can be placed after a race in thirteen different ways, thus:—A B C, A C B, B A C, B C A, C A B, C B A, or A B C as a dead heat; or A B, A C, or B C equal for the first place; or A first with B C equal seconds; or B first with A C equal seconds; or C first with A B equal seconds.

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136.

The 34 points scored against Oxbridge by the New Zealanders can be made up in two ways, either by 8 tries and 2 converted tries, or by 3 tries and 5 converted tries.

The highest possible score on these lines is 10 tries converted, equalling 50 points, and as the New Zealanders’ score, if all tries are converted, becomes four-fifths of this, their actual score was 3 tries and 5 converted into goals.

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137.

The smallest number, of which the alternate figures are cyphers, which is divisible by 9 and by 11 is 909090909090909090909!

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138.

Our problem in which it is stated that A with 8d. met B and C with five and three loaves, and asked how the cash should be divided between B and C, if all agreed to share the loaves. Now each eats two loaves and two-thirds of a loaf, and B gives seven-thirds of a loaf to A, while C gives him one-third of a loaf. So B receives 7d. and C 1d.

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139.

When, on opening four money-boxes containing pennies only, it was found that those in the first with half of all the rest, those in the second with a third of the others, those in the third with a fourth, and those in the fourth with a fifth of all the rest, amounted in each case to 740, the four boxes held £6. 1s. 8d., and the numbers of pennies were 20, 380, 500, and 560.

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140.

If two steamers, A and B, start together for a trip to a distant buoy and back, and A steams all the time at ten knots an hour, while B goes outward at eight knots and returns at twelve knots an hour, B will regain port later than A, because its loss on the outward course will not have been recovered on the run home.

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141.

If in London a new head to a golf club costs four times as much as a new leather face, while at St Andrews it costs five times as much, and if the leather face costs twice as much in London as in St Andrews, and if, including a shilling paid for a ball, the charges in London were twice as much as they would have been at St Andrews, the London cost of a new head is four shillings, and of a leather face a shilling.

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142.

When two children were asked to give the total number of sheep and cattle in a pasture, from the number of each sort, and one by subtraction answered 10, while the other arrived at 11,900 by multiplication, the true numbers were 170 sheep, 70 cattle, 240 in all.

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143.

If a man picks up one by one fifty-two stones, placed at such intervals on a straight road that the second is a yard from the first, the third 3 yards from the second, and so on with intervals increasing each time by 2 yards, and bring them all to a basket placed at the first stone, he has to travel about 52 miles, or, to be quite exact, 51 miles, 1292 yards.

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144.

When, in the House of Commons, if the Ayes had been increased by 50 from the Noes, the motion would have been carried by 5 to 3; and if the Noes had taken 60 votes from the Ayes it would have been lost by 4 to 3, the motion succeeded; 300 voted “Aye,” and 260 “No.”

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145.

There are 143 positions on the face of a watch in which the places of the hour and minute-hands can be interchanged, and still indicate a possible time. There would be 144 such positions but for the fact that at twelve o’clock the hands occupy the same place.

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146.

If in a cricket match the scores in each successive innings are a quarter less than in the preceding innings, and the side which goes in first wins by 50 runs, the complete scores of the winners are 128 and 72, and of the losers 96 and 54.

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147.

When a ball is thrown vertically upwards, and caught five seconds later, it has risen 100 feet. It takes the same time to rise as to fall, and when a body falls from rest, it travels a number of feet represented by sixteen times the square of the time in seconds.

Hence comes the rule that the height in feet of a vertical throw is found by squaring the time in seconds of its flight, and multiplying by four.

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148.

The carpet which, had it been 5 feet broader and 4 feet longer, would have contained 116 more feet, and if 4 feet broader and 5 longer 113 more, was 12 feet long and 9 feet broad.

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149.

When, in estimating the cost of a hundred similar articles, shillings were read as pounds, and pence as shillings, and the estimated cost was in consequence £212, 18s. 4d. in excess of the real cost, the true cost of each article was 2s. 5d.

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150.

If the square of the number of my house is equal to the difference of the squares of the numbers of my next door neighbours’ houses, and if my brother in the next street can say the same of his house, though its number is not the same as that of mine, our houses are numbered 8 and 4. In my street the even numbers are all on one side, in my brother’s street they, run odd and even consecutively, and so 8² = 10² - 6², and 4² = 5² - 3².

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151.

When two men of unequal strength have to move a block which weighs 270 ℔s., on a light plank 6 feet long, if the stronger man can carry 180 ℔s., the block must be placed 2 feet from him, so that he may have that share of the load.

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152.

If a man who had twenty coins, some shillings, and the rest half-crowns, were to change the half-crowns for sixpences, and the shillings for pence, and then found that he had 156 coins, he must have had eight shillings at first.

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153.

If, when coins are placed on a table at equal distances apart, so as to form sides of an equilateral triangle, and when as many are taken from the middle of each side as equal the square root of the number on that side, and placed on the opposite corner, the number on each side is then to the original number as five is to four, there are forty-five coins in all.

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154.

When the gardener found that he would have 150 too few if he set his posts a foot apart, and seventy to spare if he set them at every yard, he had 180 posts.

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155.

In order to buy with £100 a hundred animals, cows at £5, sheep at £1, and geese at 1s. each, the purchaser must secure nineteen cows, one sheep, and eighty geese.

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156.

If John, who is 21, is twice as old as Mary was when he was as old as Mary is, Mary’s age now is 15³⁄₄ years.

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157.

If in a cricket match A makes 35 runs, and C and D make respectively half and a third of B’s score, and if B scores as many less than A as C scores more than D, B made 30, C 15, and D 10 runs.

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158.

The least number which, divided by 2, 3, 4, 5, 6, 7, 8, 9, or 10, leaves remainders 1, 2, 3, 4, 5, 6, 7, 8, 9, is 2519, their least common multiple less 1.

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159.

A square table standing on four legs, which are set at the middle points of its sides, can at most uphold its own weight upon one of its corners.

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160.

The division of ninety-nine pennies, so that share 1 exceeds share 2 by 3, is less than share 3 by 10, exceeds share 4 by 9, and is less than share 5 by 16, is 17, 14, 27, 8, and 33.

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161.

If Indians carried off a third of a flock and a third of a sheep, and others took a fourth of the remainder and a fourth of a sheep, and others a fifth of the rest and three-fifths of a sheep, and there were then 409 left, the full flock was 1025 sheep.

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162.

When a cistern which held fifty-three gallons was filled by three boys, A bringing a pint every three minutes, B a quart every five minutes, and C a gallon every seven minutes, it took 230 minutes to fill it, and B poured in the final quart, A and C coming up one minute too late to contribute at the last.

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163.

A man who said, late in the last century, that his age then was the square root of the year in which he was born, was speaking in the year 1892.

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164.

If when a dealer in curios sold a vase for £119, his profit per cent., and the cost price of the vase, were expressed by the same number, it had cost him £70.

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165.

The chance of throwing at least one ace in a single throw with a pair of dice is ¹¹⁄₃₆, for there are five ways in which each dice can be thrown so as not to give an ace, so that twenty-five possible throws exclude aces. Hence the chance of not throwing an ace is ²⁵⁄₃₆, which leaves ¹¹⁄₃₆ in favour of it.

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166.

The policeman who ran after a thief starting four minutes later, and running one-third faster, if they both ran straight along the road, caught him in twelve minutes.

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167.

At a bazaar stall, where twenty-seven articles are exposed for sale, a purchaser may buy one thing or more, and the number of choices open to him is one less than the continued product of twenty-seven twos, or 134217727.

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VERY PERSONAL

168. When Nellie’s father said:—

he was 42, and she was 14.

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WORD AND LETTER PUZZLES

1.

The word square is completed thus:—

M E A D

E D G E

A G U E

D E E D

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2.

The word square filled in is:—

C I R C L E

I N U R E S

R U D E S T

C R E A S E

L E S S E E

E S T E E M

Notice the curious diagonal of E’s.

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3.

In the incomplete sentence,

SO * * * * AG * * * * LATI * * * * X * * * * ITH

the duplicate letters are filled in thus:—

SOME MEAGRE RELATIVE VEXED EDITH

The two last letters of each word are repeated as the two first of the word that follows.

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4.

The word square is completed thus:—

W A S T E

A C T O R

S T O N E

T O N I C

E R E C T

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5.

By twice building up two Ds into a B we make BULBOUS.

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6.

The question put on paper to the love-lorn youth, “Loruve?” is, when interpreted, “Are you in love?” and the advice given to him on another slip, “Prove L A FD and ensure success,” reads into, “Prove a fond lover, and ensure success” (a f on d l over).

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7. DOUBLE ACROSTIC ST. JAMES’ GAZETTE

S prin G

T ar A

J abe Z

A t E

M omen T

E mme T

S trik E

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8.

The completed word square is—

A M E N D S

M I N I O N

E N A B L E

N I B B L E

D O L L A R

S N E E R S

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9.

The oracular response to a young Frenchman at a fête, who inquired how he could best please the ladies—

MEC DO BIC

conceals this sage advice—

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10.

The solution to our Letter Fraction Problem is of a verbal character. The original statement

m ot y = mo

is dealt with thus:—

m on ot on y = mo not on y, and so the word monotony solves the equation.

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11.

The buried beasts are chamois, buffalo, heifer, and leopard; and when the Oxford athlete cries—

he introduces us to a porcupine.

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12.

The lines—

is solved by indivisibility, which has many an i, like a peacock’s tail.

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13.

The English word of thirteen letters in which the same vowel occurs four times, the same consonant six times, another twice, and another once, is Senselessness.

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14.

The condensed proverb “WE IS DO” reads at its full length as “Well begun is half done.”

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15.

This is the completed word square:—

W A S H E S

A R T E R Y

S T O R M S

H E R M I T

E R M I N E

S Y S T E M

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16.

Dr Whewell’s puzzle lines—

O O N O O.

U O A O O I O U

O N O O O O M E T O O.

U O A O I D O S O

I O N O O I O U T O O!

read thus:—

OH SIGH FOR NO CIPHER

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17.

This is the completed diamond:—

P

P O R

C O R E S

F O R C E P S

P O R C E L A I N

R E F L E C T

S P A C E

S I T

N

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18.

The medley—

Tan HE Edsa VEN in It N Gja SmeTs AsgN aD Az Rett De

is read by taking first the capitals in their order, and then the small type. It comes put out as “The Evening Standard and Saint James’s Gazette.”

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19.

The statement, I can travel first-class on the G.E.R. from 2222222244444500, reads into—from 22 to 2 to 22 to 4 for 44 4d; or, in plain terms, from 1.38 to 3.38 for 14s. 8d. This works out at about 3d. a mile, the usual allowance for first-class, for two hours, at about 29 miles an hour.

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20. A CURIOUS OLD INSCRIPTION

Read the inscription backwards, and it resolves itself into the lines familiar to us in our childhood:—

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21.

The reason why the invitation to Jack to sample the Irish stew at Simpson’s was to be kept in mind by the catch words—

Join me at and Join me at ai Join me at as

is because if you join me and at, and note that and is on i, which is on s, you arrive at the suggestive sentence—meat and onions.

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22.

The labourer’s quaint letter, which ran “Cepatomtogoatatrin,” was, in plainer English, “Kept at home to go a tatering.”

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23.

Our double acrostic comes out thus:—

Problem—Puzzles

P a P

(T) R U (E)

O Z

B o Z

L eve L

E v E

M S

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24.

The Hidden Proverb is—

“Necessity is the mother of invention.”

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25.

The “deed done” in our Will puzzle is the making in Roman numerals of “Codicil.” The lawyer was to set down, a hundred, to add nothing, to set down five hundred, then one, then another hundred, and then one more, and, finally, fifty, and accordingly he wrote upon the parchment the one word CODICIL.

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26.

The word square is—

E D I T O R

D E S I R E

I S L A N D

T I A R A S

O R N A T E

R E D S E A

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27.

The notes of music A. G. A. E. A. over the grave of a French musician, who was choked by a fish bone, are in the French notation, “La sol la mi la,” which reads into “La sole l’a mis 1à.”

Similarly the inscription over the porch of Gustave Doré’s house C. E. B. A. C. D. is equivalent to “Do, mi, si, la, do, re,” which may be taken to represent “Domicile à Doré.”

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28.

When his best girl said to Jack Spooner, “We can go to-morrow at 222222222222 LEY STREET,” he understood her to mean, “We can go to-morrow, at two minutes to two, to two twenty-two, to 222 Tooley Street.”

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29

CT T T T T T T T T T spells contents (c on ten ts!).

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30.

We can treat the word disused so as to affirm or to disallow the use of its initial or final d, for we can write it d is used, or disuse d!

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31.

The title of the book shaken up into

E I O O O U B C N N R R S S

is “Robinson Crusoe.”

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32.

If the letter M only is inserted in the proper places in the line—

A DEN I I CAN DOCK

it will read: Madmen mimic and mock.

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33.

The quotation from Shakespeare—

OXXU8 MAAULGIHCTE

NOR

is by interpretation:—“Nothing extenuate, nor set down aught in malice.”

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34.

The phonetic nightmare—

Ieukngheaurrhphthewempeighghteaps—

is merely the word unfortunates. It can be justified thus by English spelling of similar sound taken letter by letter:—

u—iew in view; n—kn in know; f—gh in tough; o—eau in beau; r—rrh in myrrh; t—phth in phthisis; u—ewe; n—mp in comptroller; a—eigh in neigh; t—ght in light; e—ea in tea; and s—ps in psalm.

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35.

The word square is completed thus:—

F A R M

A R E A

R E N T

M A T E

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36. A QUAINT INSCRIPTION

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37. A POET’S PI

is disentangled thus:—

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38. BURIED PLACES

The five buried places are Deal, London, Esk, Perth, and Baden. The word is Ourangoutang.

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39.

It is no offence to conspire in the evening, because what is treasonable is reasonable after t!

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40.

The bit of botany—

is solved by, me, an em on e, the delicately hung wind-flower.

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41. A PIED PROVERB

The pied proverb, is “A rolling stone gathers no moss.”

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42.

The Drop Letter Proverb—

E..t. .e.s..s .a.e .h. .o.. ..i.e, is—Empty vessels make the most noise.

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43.

The English word of five syllables, which has eight letters, five of them vowels—namely an a, an e, twice i, and y—is Ideality.

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44.

TORMENT may be turned into RAPTURE, using four links, changing only one letter each time, and varying the order of the letters, thus: TORMENT, portent, protest, pratest, praters, RAPTURE.

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45.

The pied sentence—

a a c e e e f f h h i i i i i m n n o o o p r r s s t t t t t

can be cast into the proverb—

“Procrastination is the thief of time.”

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46.

The English sentence, when the letter o is added, reads:—

“Good old port for orthodox Oxford dons.”

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47

is solved by Strengths.

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48.

The letters AAAAABBNNIIRSSTT form the word Antisabbatarians.

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49.

The quotation from Shakespeare,

KINI

stands for “A little more than kin, and less than kind.”

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50.

The two English words which have the first six letters of the alphabet among their ten letters are fabricated and bifurcated.

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51. SHIFTING NUMBERS

The letter A stands at the head of the letters of the alphabet. For bed 3 of these are used; for goal, 4; for prison, 6; for six, 3; for three, 5. The letter A is not used in the spelling of the name of any number from 1 to 100, but it makes up, with the other vowels, the number 6.

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52.

The prodigal’s letter to his father, “Dear Dad, keep 1000050,” in reply to a suggestion for safeguarding some of his prospects, was written in playful impudence; and its interpretation is, “Dear Dad, keep cool!” for the figures in Roman numerals are COOL.

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53. BURIED POETS

The poets’ names buried in the lines—

are Gray, Moore, Byron, Pope, Dryden, Gay, Keats, and Hemans.

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54.

When A. B. gave up the reins of government, and C.B. took office in his place, the two verbs, similar in all respects, except that the one is longer by one letter than the other, which expressed the change, were resigns—reigns.

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55

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56

is solved by Cataract. The first line reads, First a c and a t, last a c and a t, that is cat and act.

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57.

Cuba.

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58.

The Rebus T S is solved by the words tones and tans, t before one s, or t before an s.

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59.

The phonetic phrase—

INXINXIN—

is, Ink sinks in!

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A GOOD END

60. Finis (F IN IS).

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