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:—
| 2 | 1 | 5 | ) | * | 7 | * | 9 | * | ( | 1 | * | * | 2 | 1 | 5 | ) | 3 | 7 | 1 | 9 | 5 | ( | 1 | 7 | 3 | |
| * | * | * | 2 | 1 | 5 | |||||||||||||||||||||
| * | 5 | * | 9 | 1 | 5 | 6 | 9 | |||||||||||||||||||
| * | 5 | * | 5 | 1 | 5 | 0 | 5 | |||||||||||||||||||
| * | 4 | * | 6 | 4 | 5 | |||||||||||||||||||||
| * | * | * | 6 | 4 | 5 | |||||||||||||||||||||
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:—
| 3 | 4 | 5 | 3 | 4 | 5 | |||||
| * | * | 3 | 7 | |||||||
| * | * | * | * | 2 | 4 | 1 | 5 | |||
| * | * | * | * | 1 | 0 | 3 | 5 | |||
| * | * | 7 | 6 | * | 1 | 2 | 7 | 6 | 5 | |
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)1 × 3 × 4 × 6 = 45
72 - 5 × 8 × 93 × 4 × 6 + 1 = 45
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:—
| 999⁄9 = 100 | IX IX IX IX IX 100 | 9 × 9 + 9 + 99⁄9 = 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 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.
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 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.
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 | 11 | 169⁄253 | |
| 240⁄253 of 1s. = | 11 | 97⁄253 | ||
| 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 11⁄5 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:—
3570 + 148296 = 1
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.
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.)
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.
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.
130.
The Queer Division—
A third of twelve divide
By just a fifth of seven;
And you will soon decide
That this must give eleven—
is solved by LV ÷ V, or 55 ÷ 5 = 11.
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.
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.
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.
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.
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.
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.
137.
The smallest number, of which the alternate figures are cyphers, which is divisible by 9 and by 11 is 909090909090909090909!
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.
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.
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.
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.
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.
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.
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.”
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
VERY PERSONAL
168. When Nellie’s father said:—
I was twice as old as you are
The day that you were born.
You will be just what I was then
When fourteen years are gone—
he was 42, and she was 14.