Solutions

Chess Cameos

No. XXVI

Black has made the false move Kt from Q sq to Kt 3. When this is replaced, and the king is moved as the proper penalty, White mates at once with one or other of the Knights.

Back to Cameo

No. XXVII

Replace the White Kt at B 7, and a Black Pawn at K 4; then P takes P en pas. Mate.

Analysis and Proof

It can be proved that Black’s last move must have been P from K 2 to K 4, so that White may take the P en pas.

The Black King cannot have moved from any occupied square.

(The White Kt now occupies B 7.)

Nor from Kt 3 or 4, as both are now doubly guarded, so that he cannot have moved out of a check.

(The White Kt now helps to guard Kt 5.)

Nor can he have moved from K 2, as the White P on Q 6 cannot have moved to give check.

No other P can have moved.

The K P cannot have moved from K 3, became of the position of the White King.

Therefore Black’s last move must have been P from K 2 to K 4, which White can take en pas giving Mate.

Back to Cameo

No. XXVIII

B—Q 2P—R 7 B—R 5P—R 7 becomes Q P—Kt 4any White hasno move.

The way in which the B runs to earth and is shut in is most ingenious. Black with the new Q cannot anyhow give White a move.

Back to Cameo

No. XXIX

R—Kt 7, ch.K moves R—Kt 5P becomes Q R—B 5, ch.Q × R, stalemate.

Back to Cameo

No. XXX

B—Kt 8.

Back to Cameo

No. XXXI

B—R 4.

Back to Cameo

No. XXXII

B—R sq.

Back to Cameo

No. XXXIII

B—Q 4.

Back to Cameo

No. XXXIV

B—B sq.

Back to Cameo

No. XXXV

K—R 4.

Back to Cameo

No. XXXVI

Kt—Kt’s 6.

Back to Cameo

No. XXXVII

R—Kt 5Kt × R Kt—B 6Kt—B 6 ch. Kt × Kt mate

Back to Cameo

No. XXXVIII

Kt—R 7B moves Q—KB 8B returns Q—R 8 mate.

Any other move of the Kt would impede the movements of the Q.

Back to Cameo

No. XXXIX

R—R sq.B moves. Q—Kt sq.B returns. Q to Kt sq. mate.

Back to Cameo

No. XL

This beautiful problem is solved by:—

Q—Kt 6P × Q K—B 2any mates accordingly.

or

Kt—K 3 R—B 3any mates accordingly.

Back to Cameo

No. XLI

R—R 2.B × Kt Q—R sq.any Q mates.

if

B—Q sq. Kt—QB 8,any Q or Kt mates.

if

B elsewhere Kt—QB 6,any Q or Kt mates.

Back to Cameo

No. XLII

Kt—Kt 4, dis. ch.K—R 8. Q—KR 2, ch.P × Q. Kt—B 2, mate.

There are other variations.

Back to Cameo

No. XLIII

B—Kt sq.P × Kt, Q—QR7,K × Kt. Q mates.

If K × Kt, Q × P, and mates next move.

Back to Cameo

No. XLIV

K—Q 7,K moves R—Q 5,K × R Q mates.

Back to Cameo

No. XLV

R—Q 8K moves Q × P, ch.K × Q B mates.

if

B moves Q—K 7any Q mates.

Back to Cameo

No. XLVI

B—B 6K × R Q—Q 7, ch.K × Q R mates.

if

B × R R—B 6, ch.K × P Q mates.

There are other variations.

Back to Cameo

No. XLVII

Q—R 8Kt × Q Kt—B 6any mates accordingly.

Back to Cameo

No. XLVIII

B—Kt 8B—K Kt 2 Q × B, ch.Kt—K 4 Q—QR 7, mate.

if

B—B 3 Q—Q 2, ch.K moves P mates.

Back to Cameo

No. XLIX

B—Kt 2K—K 4 Q—K 3 ch.any mates.

There are other variations.

Back to Cameo

No. L

Q—KR 2K—B 3 or B4 Q—Q 6, ch.Q × Q Kt—K 5. double ch. mate.

There are other variations.

Back to Cameo

No. LI

R—QR sq.P moves R—R 2P × R P mates.

Back to Cameo

No. LII

Q—B sq.P—K 7 Q × BP ch.K—Q 3 Q mates.

if

K—Q 2 Q × BP ch.K—B 3 Q mates.

There are other variations.

Back to Cameo

No. LIII

B—R 8K—R 2 Q—QR sq.K moves Q—Q Kt. 7, mate.

Back to Cameo

No. LIV

Q—R 5B × Q or B—B 2 Kt—Kt 5any Kt mates.

Back to Cameo

No. LV

K—Kt 7Kt—B 3 Q—Q 5, ch.Kt × Q Kt. mates.

if

P—K 3 Q—B 8, ch.K moves Q mates.

Back to Cameo

No. LVI

R—Kt 6P moves B—Kt 4P × B R × P mate.

Back to Cameo

No. LVII

Kt from R 3—Kt 5P × Kt B—B 4P × B P—Q 4 ch.P × P en pas R—B 5 mate.

if

R × B R—Kt 5 ch.any mates accordingly.

and if

Q—K 6 Kt—Q 6Q—Kt 3 R—B 5 ch.Q × R Kt—B 7 mate.

Back to Cameo

Science at Play

No. LVIII.—The Geared Wheels

The arrow head at the top of a small wheel with ten teeth, which is geared into and revolved round a large fixed wheel with forty teeth, will point directly upwards five times in its course round the large wheel. Four of these turnings are due to the rotation of the small wheel on its own axis, and one of them results from its revolution round the large wheel.

Back to Description

No. LX.—The Fifteen Bridges

It is possible to pass over all the bridges which connect the islands A and B and the banks of the surrounding river without going over any of them twice.

The course can be shown thus, using capital letters for the different regions of land, and italics for the bridges:—Ea Fb Bc Fd Ae Ff Cg Ah Ci Dk Am En Ap Bq ElD.

This order of the bridges can, of course, be reversed.

Back to Description

No. LXVI.—A Duck Hunt

In order that a spaniel starting from the middle of a circular pond, and going at the same pace as a duck that is swimming round its edge, shall be sure to catch it speedily, the dog must always keep in the straight line between the duck and the centre of the pond.

The duck can never gain an advantage by turning back, and if it swims on continuously in a circle it will be overtaken when it has passed through a quarter of the circumference, for the dog will in the same time have described a semi-circle whose diameter is the radius of the pond, ending at the point where the duck is caught.

Back to Description

No. LXIX.—The Tethered Bird

When a bird tethered by a cord 50 feet long to a post 6 inches in diameter uncoils the full length of the cord, and recoils it in the opposite direction, keeping it always taut, it flies 10,157 feet, or very nearly 2 miles, in its double course.

To avoid possible misunderstanding, we point out that, in order to pass from the uncoiling to the recoiling position, the bird must fly through a semicircle at the end of the fully extended cord.

Back to Description

No. LXX.—The Moving Disc and the Fly

When a fly, starting from the point A, just outside the revolving disc, and always making straight for its mate at the point B, crosses the disc in four minutes, during which time the disc is turning twice, the revolution of the disc has a most curious and interesting effect on the path of the fly.

The fly is a quarter of a minute in passing from the outside circle to the next, during which the disc has made an eighth of a revolution, and the fly has reached the point marked 1. The succeeding points up to 16 show the position of the fly at each quarter of a minute, until, by a prettily repeated curve, B is reached.

Back to Description

No. LXXI.—A Shunting Puzzle

The following method enables the engine R to interchange the positions of the wagons, P and Q, for either of which there is room on the straight rails at A, while there is not room there for the engine, which, if it runs up either siding, must return the same way:—

1. R pushes P into A. 2. R returns, pushes Q up to P in A, couples Q to P, draws them both out to F, and then pushes them to E. 3. P is now uncoupled, R takes Q back to A, and leaves it there. 4. R returns to P, pulls P back to C, and leaves it there. 5. R, running successively through F, D, B, comes to A, draws Q out, and leaves it at B.

Back to Description

No. LXXV.—Pharaoh’s Seal

It is quite puzzling to decide how many similar triangles or pyramids are expressed on the seal of Pharaoh. There are in fact 96.

Back to Description

No. LXXVI.—Round the Garden

The four persons who started at noon from the central fountain, and walked round the four paths at the rates of two, three, four, and five miles an hour would meet for the third time at their starting point at one o’clock, if the distance on each track was one-third of a mile.

Back to Description

No. LXXVII.—A Joiner’s Puzzle

This diagram shows how to divide Fig. A into two parts, and so rearrange these that they form either Fig. B or Fig. C, without turning either of the pieces.

Cut the five steps, and shift the two pieces as is shown.

Back to Description

No. LXXVIII.—The Broken Octagon

The Broken Octagon is repaired and made perfect if its pieces are put together thus:—

Back to Description

No. LXXIX.—At A Duck Pond

The pond was doubled in size without disturbing the duck-houses, thus:—

Back to Description

No. LXXX.—All on the Square

This is a perfect arrangement:—

Back to Description

No. LXXXI.—Pins and Dots

The pins may be placed thus:—

On the third dot in the top line; on the sixth dot in the second line; on the second dot in the third line; on the fifth dot in the fourth line; on the first dot in the fifth line; on the fourth dot in the sixth line.

Back to Description

No. LXXXII.—A Tricky Course

To trace this course draw lines upon the diagram from square 46 to squares 38, 52, 55, 23, 58, 64, 8, 57, 1, 7, 42, 10, 13, 27, and 19. This gives fifteen lines which pass through every square only once.

Back to Description

No. LXXXIII.—For the Children

Make a square with three on every side, and place the remaining four one on each of the corner men or buttons.

Back to Description

No. LXXXVII.—Loyd’s Mitre Problem

The figure given is thus divided into four equal and similar parts:—

Back to Description

No. LXXXIX.—Cut Off the Corners

A very simple rule of thumb method for striking the points in the sides of a square, which will be at the angles of an octagon formed by cutting off equal corners of the square, is to place another square of equal size upon the original one, so that the centre is common to both, and the diagonal of the new square lies upon a diameter of the other parallel to its side.

Back to Description

No. XCIII.—Making Many Squares

The subjoined diagram shows how the two oblongs, applied to the two concentric squares, produce 31 perfect squares, namely, 17 small ones, one equal to 25 of these, 5 equal to 9, and 8 equal to 4.

Back to Description

No. XCIV.—Cut Across

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:—

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

No. CVIII.—Still A Square

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.

Back to Description

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.

Back to Description

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:—

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Problem

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

Back to Problem

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.

Back to Problem

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 7⁄2 seconds after it ceased, so I heard the whistle for 5 minutes, 2⁄2 seconds.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

8.

When two volumes stand in proper order on my bookshelf, each 2 inches thick over all, with covers ⁄8 of an inch in thickness, a bookworm would only have to bore ⁄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.

Back to Problem

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 14⁄2 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.

Back to Problem

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!

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.”

Back to Problem

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!

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.

Back to Problem

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.”

Back to Problem

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.

Back to Description

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.

Back to Description

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.

Back to Description

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

9.

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

Back to Description

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

Back to Description

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

Back to Description

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

7 - 5 × 8 × 93 × 4 × 6 + 1 = 45

Back to Description

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 27⁄2 years old, and Ann is 16⁄2.

For, tracing the question backwards, when Ann was 5⁄2 Mary was 16⁄2. When Ann is three times that age she will be 49⁄2. The half of this is 24⁄4, and when Mary was at that age Ann was 13⁄4. Mary’s age, by the question, was twice this, or 27⁄2.

Back to Description

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.

Back to Description

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.

Back to Description

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 3⁄2 caskfuls of wine.

Back to Description

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.

Back to Description

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.

Back to Description

19.

The fallacy of the equation—

4 - 10 = 9 - 15
4 - 10 + ⁄4 = 9 - 15 + ⁄4

and the square roots of these—

2 - ⁄2 = 3 - ⁄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 ⁄2, we have either 2 = 2, or 3 = 3.

Back to Description

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 2⁄11 cubic feet.

Back to Description

21.

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

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

Back to Description

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 1⁄8 and 9⁄8, for 1⁄8 + 9⁄8 = 11, and ⁄8 ÷ ⁄8 = 7.

Back to Description

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.

Back to Description

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.

Back to Description

38.

The puzzle—

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

is solved by fie, seen.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 8⁄5 feet.

Back to Description

43.

The hour that struck was twelve o’clock.

Back to Description

44.

Sixty years.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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!)

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 ⁄46656, and in twenty throws by about ⁄2332.

Back to Description

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.

Back to Description

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 ⁄12 wine and ⁄12 water.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 + ⁄2 = ⁄2; and ⁄2 × ⁄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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

69.

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

(8 - 5 + 1) = (11 - 9) = (7 - 3) = (6 + 2) = 4(10).

Back to Description

70.

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

Back to Description

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.

Back to Description

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.

Back to Description

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 99⁄101 and ⁄101.

Back to Description

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.

Back to Description

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.

Back to Description

76.

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

80.

He sent 163. She sent 157.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 7⁄2 inches deep.

Back to Description

93.

Aminta was eighteen.

Back to Description

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.

Back to Description

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?

Back to Description

96.

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

s. d.

⁄253 of £1 = 18 11 ⁄253

⁄253 of 1s. = 11 ⁄253

⁄253 of 1d. = ⁄253

£1 0 0

Back to Description

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.

Back to Description

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.

Back to Description

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 12⁄2, 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.

Back to Description

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, ⁄3 and ⁄3, and was able to show that ⁄3 + ⁄9 = ⁄9, and that ⁄3 + ⁄9 = ⁄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 a + b = b + a

a - b = a - b

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

therefore a + b = 1.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 41⁄13 minutes past 3.

Back to Description

109.

If the walnut monkey had stopped to help the other, and they had eaten filberts at equal rates, they would have escaped in 2⁄4 minutes.

Back to Description

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.

Back to Description

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!

Back to Description

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.

Back to Description

113.

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

Back to Description

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 1⁄5 seconds each.

Back to Description

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.

Back to Description

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!

Back to Description

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.”

Back to Description

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.

Back to Description

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 2⁄4 years, the car is now 12 months old, and the tyres have had 9 months’ wear.

Back to Description

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.

Back to Description

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.

Back to Description

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.)

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 3⁄8 miles an hour.

Back to Description

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.

Back to Description

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.

Back to Description

137.

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

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.”

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 8 = 10 - 6, and 4 = 5 - 3.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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 15⁄4 years.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

165.

The chance of throwing at least one ace in a single throw with a pair of dice is ⁄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 ⁄36, which leaves ⁄36 in favour of it.

Back to Description

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.

Back to Description

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.

Back to Description

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.

Back to Description

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

Back to Puzzle

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.

Back to Puzzle

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.

Back to Puzzle

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

Back to Puzzle

5.

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

Back to Puzzle

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).

Back to Puzzle

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

Back to Puzzle

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

Back to Puzzle

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—

Aimez, cédez, obéissez!
Love, yield, obey!

Back to Puzzle

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.

Back to Puzzle

11.

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

“Though I jump and row and run,
Cap or cup I never won”

he introduces us to a porcupine.

Back to Puzzle

12.

The lines—

Fourteen letters here we fix,
Vowels only two are spoken;
All together these we mix
Into what can not be broken—

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

Back to Puzzle

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.

Back to Puzzle

14.

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

Back to Puzzle

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

Back to Puzzle

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

You sigh for a cipher, O, I sigh for you,
Sigh for no cipher, O sigh for me too.
You sigh for a cipher, I decipher so,
I sigh for no cipher, I sigh for you too!

Back to Puzzle

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

Back to Puzzle

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.”

Back to Puzzle

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.

Back to Puzzle

20. A Curious Old Inscription

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

Ride a cock horse to Banbury Cross
To see a fine lady ride on a grey horse.
Rings on her fingers and bells on her toes,
She shall have music wherever she goes!

Back to Puzzle

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.

Back to Puzzle

22.

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

Back to Puzzle

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

Back to Puzzle

24.

The Hidden Proverb is—

“Necessity is the mother of invention.”

Back to Puzzle

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.

Back to Puzzle

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

Back to Puzzle

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.”

Back to Puzzle

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.”

Back to Puzzle

29

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

Back to Puzzle

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!

Back to Puzzle

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.”

Back to Puzzle

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.

Back to Puzzle

33.

The quotation from Shakespeare—

OXXU8 MAAULGIHCTE

NOR

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

Back to Puzzle

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.

Back to Puzzle

35.

The word square is completed thus:—

F A R M

A R E A

R E N T

M A T E

Back to Puzzle

36. A Quaint Inscription

The millers leave the mill,
The wherrymen lower their sail;
The maltsters leave the kiln,
For a drop of the White Swan’s ale.

Back to Puzzle

37. A Poet’s Pi

TONDEBNIOTOCHUMFOARYHUR
OTDIRECTTHAWHOTERSOFKLSYA;
TIKATESTUBALIGHTSTILLETRUFLYR
OTBOWLALLNFESLEAVARFWYAA

is disentangled thus:—

Don’t be in too much of a hurry
To credit what other folks say;
It takes but a slight little flurry
To blow fallen leaves far away.

Back to Puzzle

38. Buried Places

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

Back to Puzzle

39.

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

Back to Puzzle

40.

The bit of botany—

Inscribe an m above a line,
And write an e below,
This woodland flower is hung so fine
It bends when zephyrs blow—

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

Back to Puzzle

41. A Pied Proverb

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

Back to Puzzle

42.

The Drop Letter Proverb—

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

Back to Puzzle

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.

Back to Puzzle

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.

Back to Puzzle

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.”

Back to Puzzle

46.

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

“Good old port for orthodox Oxford dons.”

Back to Puzzle

47

One vowel in an English word is found,
Which by eight consonants is hedged around—

is solved by Strengths.

Back to Puzzle

48.

The letters AAAAABBNNIIRSSTT form the word Antisabbatarians.

Back to Puzzle

49.

The quotation from Shakespeare,

KINI

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

Back to Puzzle

50.

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

Back to Puzzle

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.

Back to Puzzle

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.

Back to Puzzle

53. Buried Poets

The poets’ names buried in the lines—

The sun is darting rays of gold
Upon the moor, enchanting spot;
Whose purpled heights, by Ronald loved,
Up open to his Shepherd cot.

And sundry denizens of air
Are flying, aye, each to his nest;
And eager make at such an hour
All haste to reach the mansions blest.

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

Back to Puzzle

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.

Back to Puzzle

55

Underdone mutton and onion make between you and me,
A glutton a little seedy after a capital tea.

Back to Puzzle

56

First a c and a t, last a c and a t,
With a couple of letters between,
Form a sight that our eyes are delighted to see,
Unless in their sight it is seen—

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.

Back to Puzzle

57.

Cuba.

Back to Puzzle

58.

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

Back to Puzzle

59.

The phonetic phrase—

INXINXIN—

is, Ink sinks in!

Back to Puzzle

A Good End

60. Finis (F IN IS).

Back to Puzzle