Fly on disc
When a fly, starting from the pointA, just outside the revolving disc, and always making straight for its mate at the pointB, crosses the disc in four minutes, during which time the discis 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,Bis reached.
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The following method enables the engineRto interchange the positions of the wagons,PandQ, for either of which there is room on the straight rails atA, while there is not room there for the engine, which, if it runs up either siding, must return the sameway:—
1.RpushesPintoA. 2.Rreturns, pushesQup toPinA, couplesQtoP, draws them both out toF, and then pushes them toE. 3.Pis now uncoupled,RtakesQback toA, and leaves it there. 4.Rreturns toP, pullsPback toC, and leaves it there. 5.R, running successively throughF,D,B, comes toA, drawsQout, and leaves it atB.
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It is quite puzzling to decide how many similar triangles or pyramids are expressed on the seal of Pharaoh. There are in fact 96.
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The four persons who started at noon from the central fountain, and walked round the fourpaths 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.
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This diagram shows how to divide Fig.Ainto two parts, and so rearrange these that they form either Fig.Bor Fig.C, without turning either of the pieces.
Puzzle
Cut the five steps, and shift the two pieces as is shown.
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The Broken Octagon is repaired and made perfect if its pieces are put togetherthus:—
Octagon
Octagon
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The pond was doubled in size without disturbing the duck-houses,thus:—
Duck pond
Duck pond
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This is a perfectarrangement:—
Square
Square
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The pins may be placedthus:—
On the third dot in the top line; on the sixthdot 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.
Grid
Grid
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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.
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Make a square with three on every side, and place the remaining four one on each of the corner men or buttons.
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The figure given is thus divided into four equal and similarparts:—
Square
Square
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Square
Square
A very simple rule of thumb method for striking the points in the sides of a square, which willbe 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.
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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.
Squares
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The Greek Cross can be divided by two straight cuts, so that the resulting pieces will form aperfect square when re-set, as is shown in thesefigures:—
Cross and square
Cross and square
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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.
Cross and square
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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 itis easy to arrange the other three pieces in a square of sixteen cells by lifting the three cells and dropping the two.
Nine cells
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Placed counters
It will be seen, on the subjoined diagram, howtwenty-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.
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Square
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 pointAto the cornersBandC. 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.
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The diagram below shows how the seven parts of the square can be rearranged so that they form the figure 8.
Figure 8
Figure 8
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Here is an oblong formed by piecing together two of the smaller triangles, and four of each of the otherpatterns—
Rectangle
Rectangle
Here isanother:—
Rectangle
Rectangle
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This diagram shows how every indication of the seventeen squares is broken up by the removalof seven of the asterisks which mark their corners.
Cross
Those surrounded by circles are to be removed.
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Puzzle
Puzzle
The dotted lines on the triangular figure showhow 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.
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To solve this puzzle slip the first coin or counter fromAtoD, then the others in turn fromFtoA, fromCtoF, fromHtoC, fromEtoH, fromBtoE, fromGtoB, and place the last onG. It can only be done by a sequence of this sort, in which each starting point is the finish of the next move.
Triangle
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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, asis shown in Fig. 1, and putting the two pieces together as is shown in Fig. 2.
Boards
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Here is anothersolution:—
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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.
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The train was whistling for 5 minutes. Sound travels about a mile in 5 seconds, so the first I heard of it was 5 seconds after it began. Its last sound reached me 71⁄2seconds after it ceased, so I heard the whistle for 5 minutes, 21⁄2seconds.
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These quarters were not so elastic as they are made to appear. In good truth, considering that the second man who was placed inAwas afterwards removed toI, no realsecondman was provided for at all.
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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.
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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.
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When two volumes stand in proper order on my bookshelf, each 2 inches thick over all, with covers1⁄8of an inch in thickness, a bookworm would only have to bore1⁄4of 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.
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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 outstripsthe falling man, but heovertakes and passes through his own screamin about 141⁄2miles, 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.
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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!
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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.
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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.
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Some say that a rubber tyre leaves a doublerut 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.”
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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 thehighest mew, for to his mathematical mind the Greek lettermuwas the coefficient of friction!
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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.
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Funnyboy had secretly prepared himself for the occasion by rubbing the chemical coating from the side of the box on to his boot.
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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.
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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.
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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.
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When the carpenter cut the doortoo little, he did not in factcut it enough, and he had to cut it again, so that it might fit.
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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.
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Dick’s feet will travel in round numbers nearly 16 feet further than his head, or to be exact, 15·707,960 feet.
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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.”
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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.
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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.
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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.
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This is the way to obtain eleven by adding one-third of twelve to four-fifths ofseven—
TW(EL)VE + S(EVEN) = ELEVEN
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Here is the completedsum:—
The clue is that no figure but 3, when multiplied into 215, produces 4 in the tens place.
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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 halfthat 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.
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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.
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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.
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It is the fraction6⁄9which is unchanged when turned over, and which, when taken thrice, and then divided by two becomes 1.
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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 tableshows:—
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Tom’s sum, which his mischievous neighbour rubbed almost out, is reconstructedthus:—
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Here are two other arrangements of the nine digits which produce 45, their sum; each is used onceonly:—
5 × 8 × 9 × (7 + 2)1 × 3 × 4 × 6= 45
72-5 × 8 × 93 × 4 × 6+ 1 = 45
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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 271⁄2years old, and Ann is 161⁄2.
For, tracing the question backwards, when Ann was 51⁄2Mary 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.
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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 ineleven; but there are two chances of throwing deuce-ace, the equivalent of three.
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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.
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The 21 casks, 7 full, 7 half full, and 7 empty, were shared equally by A, B, and C, asfollows:—
Thus each had 7 casks, and the equivalent of 31⁄2caskfuls of wine.
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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.
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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, therewill be in the end exactly as much lard in the butter as there is butter in the lard.
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The fallacy of theequation—
4 - 10 = 9 - 154 - 10 +25⁄4= 9 - 15 +25⁄4
and the square roots ofthese—
2 -5⁄2= 3 -5⁄2therefore 2 = 3
is explained thus:—The fallacy lies in ignoring the fact that the square roots areplus or minus. In the working we have taken both roots asplus. If we take one root plus, and the other minus, and add5⁄2, we have either 2 = 2, or 3 = 3.
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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 26⁄11cubic feet.
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We can show, or seem to show, that either four, five, or six nines amount to 100,thus:—
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This is the magic square arrangement, socontrived that the products of the rows, columns, and diagonals are all 1,000.
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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 forhalf a day.
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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.
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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.
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The difference of the ages ofAandBwho were born in 1847 and 1874, is 27, or 30 - 03. Hence, whenAwas 30Bwas 03. AndAwas 30 in 1877. Eleven years laterAwas 41 andB14, and eleven years after thatAwas 52 andB25. Thus the same two digits served to express theages of both in 1877, 1888, and 1899. This can only happen in the cases of those whose ages differ by some multiple of nine.
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