Chapter V: Hand Shadows and how to work them. Illustrated (8)

The Greek Cross can be divided by two straight cuts, so that the resulting pieces will form a perfect square when re-set, as is shown in these figures:--

No. CV.--A TRANSFORMATION

The diagram which is given below shows how the irregular Maltese Cross can be divided by two straight cuts into four pieces, which form when properly rearranged, a perfect square.

No. CVI.--SHIFTING THE CELLS

The following diagram shows by its dark lines how the whole square can be cut into four pieces, and these arranged as two perfect squares in which every semicircle still occupies the upper half of its cell.

One piece forms a square of nine cells, and it is easy to arrange the other three pieces in a square of sixteen cells by lifting the three cells and dropping the two.

+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ ∥ / \ |
| / \ | / \ | / \ | / \ ∥ / \ |
|/ \|/ \|/ \|/ \∥/ \|
| | | | ∥ |
| | | | ∥ |
+-----------+-----------+-----------+-----------+-----------+
| / \ | / \ | / \ | / \ ∥ / \ |
| / \ | / \ | / \ | / \ ∥ / \ |
|/ \|/ \|/ \|/ \∥/ \|
| | | | ∥ |
| | | | ∥ |
+-----------+-----------+===========+===========+===========+
| / \ | / \ ∥ / \ | / \ | / \ |
| / \ | / \ ∥ / \ | / \ | / \ |
|/ \|/ \∥/ \|/ \|/ \|
| | ∥ | | |
| | ∥ | | |
+-----------+===========+-----------+-----------+-----------+
| / \ ∥ / \ ∥ / \ | / \ | / \ |
| / \ ∥ / \ ∥ / \ | / \ | / \ |
|/ \∥/ \∥/ \|/ \|/ \|
| ∥ ∥ | | |
| ∥ ∥ | | |
+===========+-----------+-----------+-----------+-----------+
| / \ | / \ ∥ / \ | / \ | / \ |
| / \ | / \ ∥ / \ | / \ | / \ |
|/ \|/ \∥/ \|/ \|/ \|
| | ∥ | | |
| | ∥ | | |
+-----------+-----------+-----------+-----------+-----------+

No. CVII.--IN A TANGLE

It will be seen, on the subjoined diagram, how twenty-one counters or coins can be placed on the figure so that they fall into symmetrical design, and form thirty rows, with three in each row.

No. CVIII.--STILL A SQUARE

A
+------+------+------+
| | | |
| | | |
+------+------+------+ C
| | |
| | |
+------+------+
B

In order that a square and an additional quarter may be divided by two straight lines so that their parts, separated and then reunited, form a perfect square, lines must be drawn from the point _A_ to the corners _B_ and _C_. Draw the figure on paper, cut through these lines, and you will find that the pieces can be so reunited that they form a perfect square.

No. CIX.--A TRANSFORMATION

The diagram below shows how the seven parts of the square can be rearranged so that they form the figure 8.

No. CX.--TO MAKE AN OBLONG

Here is an oblong formed by piecing together two of the smaller triangles, and four of each of the other patterns--

Here is another:--

No. CXI.--SQUARES ON THE CROSS

This diagram shows how every indication of the seventeen squares is broken up by the removal of seven of the asterisks which mark their corners.

Those surrounded by circles are to be removed.

No. CXII.--A CHINESE PUZZLE

|\
| \
| \
| \
| \
| \
| +--\
| | \
| +--+--+ \
| | | \
| +--+ +--+\
| | | \
+--+ +--+\
| | \
+-----------------+--+

The dotted lines on the triangular figure show how a piece of cardboard cut to the shape of Fig. 1 can be divided into three pieces, and rearranged so that these form a star shaped as in Fig. 2.

No. CXIII.--FIRESIDE FUN

To solve this puzzle slip the first coin or counter from _A_ to _D_, then the others in turn from _F_ to _A_, from _C_ to _F_, from _H_ to _C_, from _E_ to _H_, from _B_ to _E_, from _G_ to _B_, and place the last on _G_. It can only be done by a sequence of this sort, in which each starting point is the finish of the next move.

AMUSING PROBLEMS

1. THE CARPENTER’S PUZZLE

The carpenter cleverly contrived to mend a hole 2 feet wide and 12 feet long, by cutting the board which was 3 feet wide and 8 feet long, as is shown in Fig. 1, and putting the two pieces together as is shown in Fig. 2.

Fig. 1.
+----------------+
| ________|
|________| |
| |
+----------------+

Fig. 2.
+------------------------+
| ________¦ |
| ¦ |
+------------------------+

2. GOLDEN PIPPINS

Here is another solution:--

1| 3| 5| 7| 9|11|13|15|17|19
39|37|35|33|31|29|27|25|23|21
2| 4| 6| 8|10|12|14|16|18|20
40|38|36|34|32|30|28|26|24|22
--+--+--+--+--+--+--+--+--+--
82|82|82|82|82|82|82|82|82|82

3. AN AWKWARD FIX

I was able to find my way in a strange district, when the sign-post lay uprooted in the ditch, without any difficulty. I simply replaced the post in its hole, so that the proper arm, with its lettering, _pointed the way that I had come_, and then, of necessity, the directions of the other arms were correct.

4. LINKED SWEETNESS LONG DRAWN OUT

The train was whistling for 5 minutes. Sound travels about a mile in 5 seconds, so the first I heard of it was 5 seconds after it began. Its last sound reached me 7¹⁄₂ seconds after it ceased, so I heard the whistle for 5 minutes, 2¹⁄₂ seconds.

5. These quarters were not so elastic as they are made to appear. In good truth, considering that the second man who was placed in _A_ was afterwards removed to _I_, no real _second_ man was provided for at all.

6. The first day of a new century can never be Sunday, Wednesday, or Friday. The cycle of the Gregorian calendar is completed in 400 years, after which all dates repeat themselves.

As in this cycle there are only four first days of a century, it is clear that three of the seven days of the week must be excluded. Any perpetual calendar shows that the four which do occur are Monday, Tuesday, Thursday, and Saturday, so that Sunday, Wednesday, and Friday are shut out.

A neat corollary to this proof is that Monday is the only day which may be the first, or which may be the last, day of a century.

7. A cricket bat with spliced handle has such good driving power, because the elasticity of the handle allows the ball to be in contact with the blade of the bat for a longer time than would otherwise be possible.

With similar effect the “follow through” of the club head at golf maintains contact with the ball, when it is already travelling fast.

8. When two volumes stand in proper order on my bookshelf, each 2 inches thick over all, with covers ¹⁄₈ of an inch in thickness, a bookworm would only have to bore ¹⁄₄ of an inch, to penetrate from the first page of Vol. I, to the last page of Vol. II, for these pages would be in actual contact if there was no binding. This very pretty and puzzling question combines in its solution all the best qualities of a clever catch with solid and simple facts.

9. A man would have to fall from a height of nearly 15 miles to reach earth before the sound of his cry as he started. The velocity of sound is constant, while that of a falling body is continually accelerated. At first the cry far outstrips the falling man, but he _overtakes and passes through his own scream_ in about 14¹⁄₂ miles, for his body falls through the 15 miles in 70 seconds, and sound travels as far in 72 seconds. Air resistance, and the fact that sound cannot pass from a rare to a dense atmosphere, are disregarded in this curious calculation.

10. A man on a perfectly smooth table in a vacuum, and where there was no friction, though no contortions of his body would avail to get away from this position, could escape from the predicament by throwing from him something which he could detach from his person, such as his watch or coat. He would himself instantly slide off in the opposite direction!

11. The monkey clinging to one end of a rope that passes over a single fixed pulley, while an equal weight hangs on the other end, cannot climb up the rope, or rise any higher from the ground.

If he continues to try to climb up, he will gradually pull the balancing weight on the other end of the rope upwards, and the slack of the rope will drop below him, while he remains in the same place.

If, after some efforts, he rests, he will sink lower and lower, until the weight reaches the pulley, because of the extra weight of rope on his side, if friction is disregarded.

12. Though the tension on a pair of traces tends as much to pull the horse backward as it does to pull the carriage forward, it is the initial pull from slack to taut which sets the traces in motion; and this, once started, must continue indefinitely until checked by a counter pull.

13. Some say that a rubber tyre leaves a double rut in dust and a single one in mud, because the air, rushing from each side into the wake of the wheel, piles up the loose dust. Others hold that the central ridge is caused by the continuous contraction of the tyre as it passes its point of contact with the road.

A correspondent, writing some years ago to “Knowledge,” said:--“It is our old friend the sucker. The tyre being round, the weight on centre of track only is great enough to enable the tyre to draw up a ridge of dust after it.”

14. If two cats on a sloping roof are on the point of slipping off, one might think that whichever had the longest paws (pause) would hold on best. Todhunter, in playful mood, saw deeper into it than that, and pronounced for the cat that had the _highest mew_, for to his mathematical mind the Greek letter _mu_ was the coefficient of friction!

15. If a penny held between finger and thumb, and released by withdrawing the finger, starts “heads” and makes half a turn in falling through the first foot, it will be “heads” again on reaching the floor, if it is held four feet above it at first.

16. HE DID IT!

Funnyboy had secretly prepared himself for the occasion by rubbing the chemical coating from the side of the box on to his boot.

17. THE CYCLE SURPRISE

If a bicycle is stationary, with one pedal at its lowest point, and that pedal is pulled backwards, while the bicycle is lightly supported, the bicycle will move backwards, and the pedal relatively to the bicycle, will move forwards. This would be quite unexpected by most people, and it is well worth trying.

18. The rough stones, by which any number of pounds, from 1 to 364, can be weighed, are respectively 1 ℔., 3 ℔s., 9 ℔s., 27 ℔s., 81 ℔s., and 243 ℔s. in weight.

19. If we disregard the resistance of the air, a small clot of mud thrown from the hindermost part of a wheel would describe a parabola, which would, in its descending limb, bring it back into kissing contact with the wheel which had rejected it.

20. THE CARELESS CARPENTER

When the carpenter cut the door _too little_, he did not in fact _cut it enough_, and he had to cut it again, so that it might fit.

21. If from the North Pole you start sailing in a south-westerly direction, and keep a straight course for twenty miles, you must steer due north to get back as quickly as possible to the Pole, if, indeed, it has been possible to start from it in any direction other than due south.

22. DICK IN A SWING

Dick’s feet will travel in round numbers nearly 16 feet further than his head, or to be exact, 15·707,960 feet.

23. A POSER

The initial letters of Turkey, Holland, England, France, Italy, Norway, Austria, Lapland, and Spain spell, and in this sense are the same as, “the finals.”

CURIOUS CALCULATIONS

1. The only sum of money which satisfies the condition that its pounds, shillings, and pence written down as a continuous number, exactly give the number of farthings which it represents, is £12, 12s., 8d., for this sum contains 12,128 farthings.

2. If, when a train, on a level track, and running all the time at 30 miles an hour, slips a carriage which is uniformly retarded by brakes, and this comes to rest in 200 yards, the train itself will then have travelled 400 yards.

The slip carriage, uniformly retarded from 30 miles an hour to no miles an hour, has an average speed of 15 miles an hour, while the train itself, running on at 30 miles an hour all the time, has just double that speed, and so covers just twice the distance.

3. The traveller had fivepence farthing when he said to the landlord, “Give me as much as I have in my hand, and I will spend sixpence with you.” After repeating this process twice he had no money left.

4. This is the way to obtain eleven by adding one-third of twelve to four-fifths of seven--

TW(EL)VE + S(EVEN) = ELEVEN

5. Here is the completed sum:--

215)*7*9*(1** 215)37195(173
*** 215
---- ----
*5*9 1569
*5*5 1505
---- ----
*4* 645
*** 645
=== ===

The clue is that no figure but 3, when multiplied into 215, produces 4 in the tens place.

6. If I attempt to buy as many heads of asparagus as can be encircled by a string 2 feet long for double the price paid for as many as half that length will encompass, I shall not succeed. A circle double of another in circumference is also double in diameter, and its area is four times that of the other.

7. If, when you reverse me, and my square, and my cube, and my fourth power, you find that no changes have been made, I am 11, my square is 121, my cube 1331, and my fourth power 14641.

8. A thousand pounds can be stored in ten sealed bags, so that any sum in pounds up to £1,000 can be paid without breaking any of the seals, by placing in the bags 1, 2, 4, 8, 16, 32, 64, 128, 256, and 489 sovereigns.

9. It is the fraction ⁶⁄₉ which is unchanged when turned over, and which, when taken thrice, and then divided by two becomes 1.

10. When the three gamblers agreed that the loser should always double the sum of money that the other two had before them, and they each lost once, and fulfilled the conditions, remaining each with eight sovereigns in hand, they had started with £13, £7, and £4 as the following table shows:--

A B C
£ £ £
At starts 13 7 4
When A loses 2 14 8
When B loses 4 4 16
When C loses 8 8 8

11. Tom’s sum, which his mischievous neighbour rubbed almost out, is reconstructed thus:--

345 345
** 37
----- -----
**** 2415
**** 1035
----- -----
**76* 12765

12. Here are two other arrangements of the nine digits which produce 45, their sum; each is used once only:--

5 × 8 × 9 × (7 + 2)
------------------- = 45
1 × 3 × 4 × 6

7² - 5 × 8 × 9
-------------- + 1 = 45
3 × 4 × 6

13. If, when the combined ages of Mary and Ann are 44, Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann, Mary is 27¹⁄₂ years old, and Ann is 16¹⁄₂.

For, tracing the question backwards, when Ann was 5¹⁄₂ Mary was 16¹⁄₂. When Ann is three times that age she will be 49¹⁄₂. The half of this is 24³⁄₄, and when Mary was at that age Ann was 13³⁄₄. Mary’s age, by the question, was twice this, or 27¹⁄₂.

14. It is safer at backgammon to leave a blot in the tables which can be taken by an ace than one which a three would hit. In either the case of an actual ace or a three the chance is one in eleven; but there are two chances of throwing deuce-ace, the equivalent of three.

15. If I start from a bay, where the needle points due north, 1200 miles from the North Pole, and the course is perfectly clear, I can never reach it if I steam continuously 20 miles an hour, steering always north by the compass needle. After about 200 miles I come upon the Magnetic Pole, which so affects the needle that it no longer leads me northward, and I may have to steer south by it to reach the geographical Pole.

16. The 21 casks, 7 full, 7 half full, and 7 empty, were shared equally by A, B, and C, as follows:--

Full cask. Half full. Empty.
A 2 3 2
B 2 3 2
C 3 1 3

or--

A 3 1 3
B 3 1 3
C 1 5 1

Thus each had 7 casks, and the equivalent of 3¹⁄₂ caskfuls of wine.

17. The foraging mouse, able to carry home three ears at a time from a box full of ears of corn, could not add more than fourteen ears of corn to its store in fourteen journeys, for it had each time to carry along two ears of its own.

18. If, with equal quantities of butter and lard, a small piece of butter is taken and mixed into all the lard, and if then a piece of this blend of similar size is put back into the butter, there will be in the end exactly as much lard in the butter as there is butter in the lard.

19. The fallacy of the equation--

4 - 10 = 9 - 15

4 - 10 + ²⁵⁄₄ = 9 - 15 + ²⁵⁄₄

and the square roots of these--

2 - ⁵⁄₂ = 3 - ⁵⁄₂

therefore 2 = 3

is explained thus:--The fallacy lies in ignoring the fact that the square roots are _plus or minus_. In the working we have taken both roots as _plus_. If we take one root plus, and the other minus, and add ⁵⁄₂, we have either 2 = 2, or 3 = 3.

20. The largest possible parcel which can be sent through the post under the official limits of 3 feet 6 inches in length, and 6 feet in length and girth combined, is a cylinder 2 feet long and 4 feet in circumference, the cubic contents of which are 2⁶⁄₁₁ cubic feet.

21. We can show, or seem to show, that either four, five, or six nines amount to 100, thus:--

IX
IX
99⁹⁄₉ = 100 IX 9 × 9 + 9 + 9⁹⁄₉ = 100
IX
IX
---
100

22. This is the magic square arrangement, so contrived that the products of the rows, columns, and diagonals are all 1,000.

+---+---+---+
| 50| 1| 20|
+---+---+---+
| 4| 10| 25|
+---+---+---+
| 5|100| 2|
+---+---+---+

23. If seven boys caught four crabs in the rock-pools at Beachy Head in six days, the twenty-one boys who searched under the seaweed and only caught one crab with the same rate of success were only at work for _half a day_.

24. A watch could be set of a different trio from a company of fifteen soldiers for 455 nights, and one of them, John Pipeclay, could be included ninety-one times.

25. If Augustus Cæsar was born September 23, B.C. 63, he celebrated his sixty-third birthday on September 23, B.C. 0; or, writing it otherwise, September 23, A.D. 0; or again, if we wish to include both symbols, B.C. 0 A.D. It is clear that his sixty-second birthday fell on September 23, B.C. 1, and his sixty-fourth on September 23, A.D. 1, so that the intervening year may be written as above.

26. The difference of the ages of _A_ and _B_ who were born in 1847 and 1874, is 27, or 30 - 03. Hence, when _A_ was 30 _B_ was 03. And _A_ was 30 in 1877. Eleven years later _A_ was 41 and _B_ 14, and eleven years after that _A_ was 52 and _B_ 25. Thus the same two digits served to express the ages of both in 1877, 1888, and 1899. This can only happen in the cases of those whose ages differ by some multiple of nine.

27 A hundred and one by fifty divide,
To this let a cypher be duly applied;
And when the result you can rightly divine,
You find that its value is just one in nine--

is solved by CLIO, one of the nine Muses.

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.

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.

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 36 12-13 minutes past 4 when I first looked. At that time the hour-hand would be pointing to 23 1-13 minutes on the dial, and at 23 1-13 minutes past 7 the hour hand would be pointing to 36 12-13 minutes.

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.

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.

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.

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.

35. The two numbers which solve the problem--

Two numbers seek which make eleven,
Divide the larger by the less,
The quotient is exactly seven,
As all who find them will confess--

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

36. There must be nine things of each sort, in order that ^99999999999999999999^ different selections may be made from twenty sorts of things.

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.

38. The puzzle--

Take five from five, oh, that is mean!
Take five from seven, and this is seen--

is solved by _fie_, _seen_.

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.

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.

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.

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.

43. The hour that struck was twelve o’clock.

44. Sixty years.

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.

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.

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.

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.

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.

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!)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

66. We can gather from the lines--

Old General Host
A battle lost,
And reckoned on a hissing,
When he saw plain
What men were slain,
And prisoners, and missing.

To his dismay
He learned next day
What havoc war had wrought;
He had, at most,
But half his host
Plus ten times three, six, ought.

One-eighth were lain
On beds of pain,
With hundreds six beside;
One-fifth were dead,
Captives, or fled,
Lost in grim warfare’s tide.

Now, if you can,
Tell me, my man,
What troops the general numbered,
When on that night
Before the fight
The deadly cannon slumbered?

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

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.

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.

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).

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

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.

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.

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.

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.

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.

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

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.

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.

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.

80. He sent 163. She sent 157.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

93. Aminta was eighteen.

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.

95. Eight and a quarter is the answer to the nonsense question--

If five times four are thirty-three,
What will the fourth of twenty be?

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 11¹⁶⁹⁄₂₅₃
²⁴⁰⁄₂₅₃ of 1s. = 11⁹⁷⁄₂₅₃
²⁴⁰⁄₂₅₃ of 1d. = ²⁴⁰⁄₂₅₃
---------------
£1 0 0
========

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.

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.

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.

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.

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.

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.

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.

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.

105. When the lady replied to a question as to her age--

If first my age is multiplied by three,
And then of that two-sevenths tripled be,
The square root of two-ninths of this is four;
Now tell my age, or never see me more--

she was 28 years old.

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.

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.

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.

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.

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.

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!

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.

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.

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 1¹⁄₁₅ seconds each.

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.

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_!

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.”

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.

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:--

35 148
-- + --- = 1.
70 296

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.

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.

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.

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.

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.

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.

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.