SCIENCE AT PLAY

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

The question, “How many times can 19 be subtracted from a million?” was set by an examiner, who no doubt expected that the answer would be obtained by dividing a million by 19. One bright youth, however, filled a neatly-written page with repetitions of

and added at the foot of the page, “N.B.—I can do this as often as you like.”

There was a touch of unintended humour in this, for, after all, the boy gave a correct answer to a badly worded question.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Here is a very smart and very simple method of dividing any multiple of 9 by 9, from the fertile brain of Lewis Carroll:—Place a cypher over the final figure, subtract the final figure from this, place the result above in the tens place, subtract the original tens figure from this, and so on to the end. Then the top line, excluding the intruded cypher, gives the result desired.Thus:—

36459 ÷ 9 = 4051,036459= 4051.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Here is one of the methods by which we can readily discover a number that is thought of. The thought-reader gives these directions to his subject: “Add 1 to three times the number you have thought of; multiply the sum by 3; add to this the number thought of; subtract 3, and tell me the remainder.” This is always ten times the number thought of. Thus, if 6 is thought of—6 × 3 + 1 = 19; 19 × 3 = 57; 57 + 6 - 3 = 60, and 60 ÷ 10 = 6.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Here is a curious rough rule for remembering distances andsizes:—

The diameter of the earth multiplied by 108 gives approximately the sun’s diameter. The diameter of the sun multiplied by 108 gives the mean distance of the earth from the sun. The diameter of the moon multiplied by 108 gives the mean distance of the moon from the earth.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Says Giles, “My wife and I are two,Yet faith I know not why, sir.”Quoth Jack, “You’re ten, if I speak true,She’s one, and you’re a cypher!”

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Here is a curious and quite uncommon method of dividing any multiple of 11 by 11.

Set down the multiple of 11, place a cypher under its last figure, draw a line, and subtract, placing the first remainder under the tens place. Subtract this from the next number in order, and so on throughout, adding in always any number that is carried.Thus:—

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

Perhaps the old saying, “there is luck in odd numbers,” may have some connection with the curious fact that the sum of any quantity of consecutive odd numbers, beginning always with 1, is the square of that number.Thus:—

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

In the number 142857, if the digits which belong to it are in succession transposed from the first place to the end, the result is in each case a multiple of the original number.Thus:—

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

By the following simple method, a plausible attempt is made to prove that 1 is equal to2:—

Suppose thata=b, then

This process only proves in reality that 0 × 1 = 0 × 2, which is true.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in four moves.

Solution

Few people know a very singular but simple method of calculating rapidly how much any given number of pence a day amounts to in a year. The rule is this:—Set down the given number of pence as pounds; under this place its half, and under that the result of the number of original pence multiplied always by five. Take, for example, 7d aday:—

The reason for this is evident as soon as we remember that the 365 days of a year may be split up into 240, 120, and 5, and that 240 happens to be the number of pence in a pound.

A small wheel with ten teeth is geared into a large fixed wheel which has forty teeth. This small wheel, with an arrow mark on its highest cog, is revolved completely round the large wheel. How often during its course is the arrow pointing directly upwards? Here is a diagram of the starting position.

Cogwheels

Solution

Here is a most curious and interesting question:—When an engine is drawing a train at full speed from York to London, what part of the train at any given moment is movingtowards York?

At any time, when the engine is drawing a train at full speed from York to London, that part of the flange of each wheel which is for the moment at its lowest is actuallymoving backwards towards York.

Cogwheels

For any point, such as A, on the circumference of the tyre, describes in running along a series of curves, as shown byfulllines in the diagram; and any point, B, on the outer edge of the flange, follows a path shown by thedottedcurves.

If these lines are followed round with a pencil in the direction of the arrows, it will be found that the point on the flange actually movesbackwardsas it passesbelow the track, while the point A, as it completes each curve, isat restfor the instant on the track, just before it starts afresh. The speed of the train does not affect these very curious facts.

In the subjoined diagram A and B represent two islands, round which a river runs as is indicated, with fifteen connecting bridges, that lead from the islands to the river’s banks.

Cogwheels

Can you contrive to pass in turn over all these bridges without ever passing over the same one twice?

Solution

In a school where two boys were taught to think out the bearings of their work, a sharp pupil remarked that 100 is represented on paper by the smallest digit and two cyphers, which are in themselves symbols of nothing. The master, quick to catch any signs of mental activity, took the opportunity to propound to his class the following ingenious puzzle:—How can the sum of 100 be represented exactly in figures and signs by making use of all the nine digits in their reverse order? This is how it isdone:—

9 × 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 100.

Another ingenious method of using the nine digits, so that by simple addition they sum up to exactly 100, and each is used once only, isthis:—

15 + 36 + 47 = 98 + 2 = 100

Here is another arrangement by which the nine digits written in their inverse order can be made to represent exactly100:—

98 - 76 + 54 + 3 + 21 = 100.

Here is yet another way of arriving at 100 by using each of the digits, this time with an0:—

401⁄25938⁄76100

Here is quite a pretty scientific experiment, which any one of a handy turn can construct andarrange:—

Looping

The spiral track is formed of two wires bent, and connected by curved cross-pieces. The upper twist is turned so that the ball starts on a horizontal course.

During the accelerated descent the ball acquires momentum enough to keep it on the vertical track, held outwardly against the wires by centrifugal force.

Convenient proportions are: height of spiral two feet, diameter six inches, and wire rails three-quarters of an inch apart.

A close approach to an ideal flying machine can be made with a little ingenuity. Two Y-shaped standards, secured to the backbone rod, support two wires which carry wings of thin silk, provided with light stays, and connected at their inner corners with the backbone by threads.

Aircraft

Rubber bands are attached to a loop on the inner end of the crank shaft, and secured to a post at the rear. These are twisted by turning the shaft with the cross wire, and when the tension is released the wings beat the air and carry the bird forward. It is known as Penaud’s mechanical bird, and has been sold as an attractive toy.

A simple apparatus constructed on the lines of this illustration will give an interesting proof of the laws which govern falling bodies on an inclined plane or on a curved path.

Apparatus

In the case of the inclined plane the ball is governed by the usual law which controls falling bodies. In that of the concave circular curve, as it is accelerated rapidly at the start, it makes its longer journey in quicker time. In the case of the cycloidal curve it acquires a high velocity. This curve has therefore been called “the curve of swiftest descent,” as a falling body passes over it in less time than upon any path except the vertical.

Here is another very simple and pretty illustration of the natural forces which come into play in “looping the loop.”

Railway

This scientific toy on a small scale may be easily made, if care is taken that the height of the higher end of the rails is to the height of the circular part in a greater ratio than 5 to 4.

A ball started at the higher end follows the track throughout, and at one point is held by centrifugal force against the under side of the rails, against the force of gravity.

If a ball is fired point blank from a perfectly horizontal gun, and travels half a mile over a level plain before it touches ground, and another similar ball is at the same moment dropped from the same height by some mechanical means, the two balls will touch ground simultaneously. The flight, however long, of one through the air has no influence upon the force of gravity, which draws it earthward at the same resistless rate as it draws the other that is merely dropped.

Cannon balls

A quick method of multiplying any number of figures by 5 is to divide them by 2, annexing a cypher to the result when there is no remainder, and if there is any remainder annexing a 5.Thus:—

464 × 5 = 2320; 464 ÷ 2 = 232, annex 0, = 2320.753 × 5 = 3765; 753 ÷ 2 = 376, annex 5, = 3765.

A duck begins to swim round the edge of a circular pond, and at the same moment a water spaniel starts from the middle of the pond in pursuit of it.

Duck and dog

If both swim at the same pace, how must the dog steer his course so that he is sure in any case to overtake the duck speedily?

Solution

Louis Napoleon, Emperor 1852

Thus, by a most remarkable series of coincidences, the principal dates of the Emperor and Empress of the French added, as is shown above, to the year of the Emperor’s accession, express in each instance the year before his fall.

In this domino diagram we have a pretty and practical proof that the squares of the sides containing the right angle in any right-angled triangle are together equal to the square of the side opposite to the right angle.

Dominoes

Each stone forms two squares, and it is easily seen that the number of squares which make up the whole square on the line opposite the right angle are equal to the number of those which make up the two whole squares on the lines which contain that angle.

A second point to be noticed is that the number of pips on the large square are equal to the number on the other two squares combined, an arrangement of the stones which forms quite a game of patience to reproduce, if this pattern is not at hand.

Four colours at most are needed to distinguish the surfaces of separate districts on any plane map, so that no two with a common boundary are tinted alike.

Map colours

On this diagram A, B, and C, are adjoining districts, on a plane surface, and X borders, in one way or another, upon each.

It is clearly impossible to introduce a fifth area which shall so adjoin these four districts as to need another tint.

Here is another freak offigures:—

A bird made fast to a pole six inches in diameter by a cord fifty feet long, in its flight first uncoils the cord,keeping it always taut, and then recoils it in the reverse direction, rewinding the coils close together. If it starts with the cord fully coiled, and continues its flight until it brings up against the pole, how far does it fly in its double course?

Tethered bird

Solution

Ask a person to write down in a line any number of figures, then to add them all together as units, and to subtract the result from the sum set down. Let him then strike out any one figure, and add the others together as units, telling you the result.

If this has been correctly done, the figure struck out can always be determined by deducting the final total from the multiple of 9 next above it. If the total happens to be a multiple of 9, then a 9 was struck out.

A fly, starting from the point A, just outside a revolving disc, and always making straight for its mate at the point B, crosses the disc in four minutes, while the disc is revolving twice. What effect has the revolution of the disc on the path of the fly?

Flies and disc

Solution

This Magic Square is so arranged that the product of the continued multiplication of the numbers in each row, column, or diagonal is 4096, which is the cube of the central 16.

The railway, D E F, has two sidings, D B A and F C A, connected at A. The rails at A, common to both, are long enough to hold a single wagon such as P or Q, but too short to admit the whole of the engine R, which, if it runs up either siding, must return the same way.

Railways

How can the engine R be used to interchange the wagons P and Q without allowing any flying shunts?—FromBall’s Mathematical Recreations.

Solution

It is a little known and very interesting fact that an equilateral triangle can easily be drawn by rule of thumb in the following way:—Take a triangle of any shape or size, and on each of its sides erect an equilateral triangle. Find and join the centres of these, and a fourth equilateral triangle is always thus formed, as shown by the dotted lines.

Triangles

These centres arecentres of gravity, and they are symmetrically distributed around the centre of gravity of the original triangle.

The figure formed by joining them must therefore be symmetrical, and, as in this case, it is a triangle, itmust bealways equilateral.

There can be no better instance of how the eye may be deceived than is so strikingly afforded in these very curiousdiagrams:—

Deception

The square which obviously contains sixty-four small squares, is to be cut into four parts, as is shown by the thicker lines. When these four pieces are quite simply put together, as shown in the second figure, there seem to be sixty-five squares instead of sixty-four.

This phenomena is due to the fact that the edges of the four pieces, which lie along the diagonal A B, do not exactly coincide in direction. In reality theyinclude a very narrow diamond, not easily detected, whose area is just equal to that of one of the sixty-four small squares.

Very curious are the results when the nine digits in reverse order are multiplied by 9 and its multiples up to 81.Thus:—

It will be seen that the figures by which the reversed digits are multiplied reappear at the beginning and end of each result except the first, and that the figures repeated between them are to be found by dividing the divisors by 9 and subtracting the result from 9. Thus, 54 ÷ 9 = 6, and 9 - 6 = 3.

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