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.
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.
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.
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.
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.
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.
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.
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.
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.
No. LXXVIII.—THE BROKEN OCTAGON
The Broken Octagon is repaired and made perfect if its pieces are put together thus:—
No. LXXIX.—AT A DUCK POND
The pond was doubled in size without disturbing the duck-houses, thus:—
No. LXXX.—ALL ON THE SQUARE
This is a perfect arrangement:—
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.
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.
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.
No. LXXXVII.—LOYD’S MITRE PROBLEM
The figure given is thus divided into four equal and similar parts:—
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.
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.
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:—
No. CV.—A TRANSFORMATION
The diagram which is given below shows how the irregular Maltese Cross can be divided by two straight cuts into four pieces, which form when properly rearranged, a perfect square.
No. CVI.—SHIFTING THE CELLS
The following diagram shows by its dark lines how the whole square can be cut into four pieces, and these arranged as two perfect squares in which every semicircle still occupies the upper half of its cell.
One piece forms a square of nine cells, and it is easy to arrange the other three pieces in a square of sixteen cells by lifting the three cells and dropping the two.
No. CVII.—IN A TANGLE
It will be seen, on the subjoined diagram, how twenty-one counters or coins can be placed on the figure so that they fall into symmetrical design, and form thirty rows, with three in each row.
No. CVIII.—STILL A SQUARE
In order that a square and an additional quarter may be divided by two straight lines so that their parts, separated and then reunited, form a perfect square, lines must be drawn from the point A to the corners B and C. Draw the figure on paper, cut through these lines, and you will find that the pieces can be so reunited that they form a perfect square.
No. CIX.—A TRANSFORMATION
The diagram below shows how the seven parts of the square can be rearranged so that they form the figure 8.
No. CX.—TO MAKE AN OBLONG
Here is an oblong formed by piecing together two of the smaller triangles, and four of each of the other patterns—
Here is another:—
No. CXI.—SQUARES ON THE CROSS
This diagram shows how every indication of the seventeen squares is broken up by the removal of seven of the asterisks which mark their corners.
Those surrounded by circles are to be removed.
No. CXII.—A CHINESE PUZZLE
The dotted lines on the triangular figure show how a piece of cardboard cut to the shape of Fig. 1 can be divided into three pieces, and rearranged so that these form a star shaped as in Fig. 2.
No. CXIII.—FIRESIDE FUN
To solve this puzzle slip the first coin or counter from A to D, then the others in turn from F to A, from C to F, from H to C, from E to H, from B to E, from G to B, and place the last on G. It can only be done by a sequence of this sort, in which each starting point is the finish of the next move.
AMUSING PROBLEMS
1. THE CARPENTER’S PUZZLE
The carpenter cleverly contrived to mend a hole 2 feet wide and 12 feet long, by cutting the board which was 3 feet wide and 8 feet long, as is shown in Fig. 1, and putting the two pieces together as is shown in Fig. 2.
2. GOLDEN PIPPINS
Here is another solution:—
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 |
| 39 | 37 | 35 | 33 | 31 | 29 | 27 | 25 | 23 | 21 |
| 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
| 40 | 38 | 36 | 34 | 32 | 30 | 28 | 26 | 24 | 22 |
| 82 | 82 | 82 | 82 | 82 | 82 | 82 | 82 | 82 | 82 |
3. AN AWKWARD FIX
I was able to find my way in a strange district, when the sign-post lay uprooted in the ditch, without any difficulty. I simply replaced the post in its hole, so that the proper arm, with its lettering, pointed the way that I had come, and then, of necessity, the directions of the other arms were correct.
4. LINKED SWEETNESS LONG DRAWN OUT
The train was whistling for 5 minutes. Sound travels about a mile in 5 seconds, so the first I heard of it was 5 seconds after it began. Its last sound reached me 71⁄2 seconds after it ceased, so I heard the whistle for 5 minutes, 21⁄2 seconds.
5.
These quarters were not so elastic as they are made to appear. In good truth, considering that the second man who was placed in A was afterwards removed to I, no real second man was provided for at all.
6.
The first day of a new century can never be Sunday, Wednesday, or Friday. The cycle of the Gregorian calendar is completed in 400 years, after which all dates repeat themselves.
As in this cycle there are only four first days of a century, it is clear that three of the seven days of the week must be excluded. Any perpetual calendar shows that the four which do occur are Monday, Tuesday, Thursday, and Saturday, so that Sunday, Wednesday, and Friday are shut out.
A neat corollary to this proof is that Monday is the only day which may be the first, or which may be the last, day of a century.
7.
A cricket bat with spliced handle has such good driving power, because the elasticity of the handle allows the ball to be in contact with the blade of the bat for a longer time than would otherwise be possible.
With similar effect the “follow through” of the club head at golf maintains contact with the ball, when it is already travelling fast.
8.
When two volumes stand in proper order on my bookshelf, each 2 inches thick over all, with covers 1⁄8 of an inch in thickness, a bookworm would only have to bore 1⁄4 of an inch, to penetrate from the first page of Vol. I, to the last page of Vol. II, for these pages would be in actual contact if there was no binding. This very pretty and puzzling question combines in its solution all the best qualities of a clever catch with solid and simple facts.
9.
A man would have to fall from a height of nearly 15 miles to reach earth before the sound of his cry as he started. The velocity of sound is constant, while that of a falling body is continually accelerated. At first the cry far outstrips the falling man, but he overtakes and passes through his own scream in about 141⁄2 miles, for his body falls through the 15 miles in 70 seconds, and sound travels as far in 72 seconds. Air resistance, and the fact that sound cannot pass from a rare to a dense atmosphere, are disregarded in this curious calculation.
10.
A man on a perfectly smooth table in a vacuum, and where there was no friction, though no contortions of his body would avail to get away from this position, could escape from the predicament by throwing from him something which he could detach from his person, such as his watch or coat. He would himself instantly slide off in the opposite direction!
11.
The monkey clinging to one end of a rope that passes over a single fixed pulley, while an equal weight hangs on the other end, cannot climb up the rope, or rise any higher from the ground.
If he continues to try to climb up, he will gradually pull the balancing weight on the other end of the rope upwards, and the slack of the rope will drop below him, while he remains in the same place.
If, after some efforts, he rests, he will sink lower and lower, until the weight reaches the pulley, because of the extra weight of rope on his side, if friction is disregarded.
12.
Though the tension on a pair of traces tends as much to pull the horse backward as it does to pull the carriage forward, it is the initial pull from slack to taut which sets the traces in motion; and this, once started, must continue indefinitely until checked by a counter pull.
13.
Some say that a rubber tyre leaves a double rut in dust and a single one in mud, because the air, rushing from each side into the wake of the wheel, piles up the loose dust. Others hold that the central ridge is caused by the continuous contraction of the tyre as it passes its point of contact with the road.
A correspondent, writing some years ago to “Knowledge,” said:—“It is our old friend the sucker. The tyre being round, the weight on centre of track only is great enough to enable the tyre to draw up a ridge of dust after it.”
14.
If two cats on a sloping roof are on the point of slipping off, one might think that whichever had the longest paws (pause) would hold on best. Todhunter, in playful mood, saw deeper into it than that, and pronounced for the cat that had the highest mew, for to his mathematical mind the Greek letter mu was the coefficient of friction!
15.
If a penny held between finger and thumb, and released by withdrawing the finger, starts “heads” and makes half a turn in falling through the first foot, it will be “heads” again on reaching the floor, if it is held four feet above it at first.
16. HE DID IT!
Funnyboy had secretly prepared himself for the occasion by rubbing the chemical coating from the side of the box on to his boot.
17. THE CYCLE SURPRISE
If a bicycle is stationary, with one pedal at its lowest point, and that pedal is pulled backwards, while the bicycle is lightly supported, the bicycle will move backwards, and the pedal relatively to the bicycle, will move forwards. This would be quite unexpected by most people, and it is well worth trying.
18.
The rough stones, by which any number of pounds, from 1 to 364, can be weighed, are respectively 1 ℔., 3 ℔s., 9 ℔s., 27 ℔s., 81 ℔s., and 243 ℔s. in weight.
19.
If we disregard the resistance of the air, a small clot of mud thrown from the hindermost part of a wheel would describe a parabola, which would, in its descending limb, bring it back into kissing contact with the wheel which had rejected it.
20. THE CARELESS CARPENTER
When the carpenter cut the door too little, he did not in fact cut it enough, and he had to cut it again, so that it might fit.
21.
If from the North Pole you start sailing in a south-westerly direction, and keep a straight course for twenty miles, you must steer due north to get back as quickly as possible to the Pole, if, indeed, it has been possible to start from it in any direction other than due south.
22. DICK IN A SWING
Dick’s feet will travel in round numbers nearly 16 feet further than his head, or to be exact, 15·707,960 feet.
23. A POSER
The initial letters of Turkey, Holland, England, France, Italy, Norway, Austria, Lapland, and Spain spell, and in this sense are the same as, “the finals.”