FREAKS OF FIGURES

Bricks

The central brick, drawn to show all its edges, as though it were made of glass, will assume the form indicated by one or other of the smaller bricks at its right and left, according to the way in which the eyes accommodate themselves for the moment to one pattern or to the other. If you do not see this at first, look steadily for awhile at the pattern you desire.

13. Why does a rubber tyre leave a double rut in dust, and a single one in mud?

Solution

14. If two cats, on opposite sides of a sharply sloping roof, are on the point of slipping off, which will hold on the longest?

Solution

If you place four coins in the positions shown at the top of this diagram, and attempt, or challenge some one to attempt, without any measuring, to move the single coin down in a straight line until the spaces from C to D on either side exactly equal the distance from A toB—

Coins

It must drop as far as is shown here, which seems to the unaided eye to be too far.

This excellent illusion can be shown as an after-dinner trick with four napkin-rings.

Here is another excellent optical illusion. Look attentively at the diagram below, and notice in which direction you apparently look into it, as though it were an open cask.

Illusion

Now shake the paper, or move it slightly, and you will find, more often than not, that you seem to see into it in quite the opposite direction.

15. I hold a penny level between my finger and thumb, and presently let it fall from the thumb by withdrawing my finger. It makes exactly a half-turn in falling through the first foot. If it starts “heads,” how far must it fall to bring it “heads” to the floor?

Solution

16. “They call these safety matches,” said Funnyboy at his club one day, “and say that they strike only on the box. Don’t believe it! I can strike them quite easily on my boot.”

No sooner said than done. He took out a match, struck it on his boot, and—phiz!—it was instantly alight. The box was handed round, and match after match was struck by the bystanders on their boots, but not one of them could succeed.

“You don’t give the magic touch,” said Funnyboy, as he gaily struck another. How did he do it?

Solution

Illusion

How many cubes can you see as you look at the large diagram? The two smaller ones should be looked at first alternately, and they will assist the eye to see at one time six, and at another time seven, very distinct cubes.

This curious optical illusion is not easily followed by eye to the finish of the several lines.

Illusion

Each short line is, in fact, part of the circumference of a circle, and the circles when completed will be found to be accurately concentric. It would seem at first sight that the lines are taking courses which would eventually meet at some point common to them all.

17. We commend this curious point to the special attention ofcyclists:—

A bicycle is stationary, with one pedal at its lowest point. If this bicycle is lightly supported, and the bottom pedal is pulled backward, what will happen?

Solution

Illusion

A most remarkable optical illusion is produced by the blending of the dark and light converging rays of this diagram. Stand with your back to the light, hold the page, or better still, the diagram copied on a card, by the lower right-hand corner, give it a continuous revolving movement in either direction, and the visible ghost of a silver coin, sometimes as large as sixpence, sometimes as large as a shilling, will appear! Where can it come from?

18. A merchant has a large pair of scales, but he has lost his weights, and cannot at the moment replace them. A neighbour sends him six rough stones, assuring him that with them he can weigh any number of pounds, from 1 to 364. What did each stone weigh?

Solution

Here is a pretty form of our firstillusion:—

Girl and goose

Place the edge of a card on the dotted line, look down upon it in a good light, and, as you drop your face till it almost touches the card, you will see the goosemove towards the sugarin the little maiden’s hand.

19. A wheel is running along a level road, and a small clot of mud is thrown from the hindermost part of the rim. What happens to it? Does it ever renew its acquaintance with the wheel that has thus rejected it?

Solution

Here is another method by which an optical illusion of length is very plainlyshown:—

Illusion

Judged by appearances, the line A B in the larger figure is considerably longer than the line A B below it, but tested by measurement they are exactly equal.

20. A village carpenter undertook to make a cupboard door. When he began to put it in its place it was too big, so he took it back to his workshop to alter it. Unfortunately he now cut it too little. What could he do? He determined to cut it again, and it at once became a good fit. How was this done?

Solution

Here is another excellent illustration that seeing is not always believing.

Illusion

No one could suppose at first sight that these four lines are perfectly straight and parallel, but they will stand the test of a straight edge. The divergent rays distract the vision.

21. If from the North Pole you start sailing in a south-westerly direction, and keep a straight course for twenty miles, to what point of the compass must you steer to get back as quickly as possible to the Pole?

Solution

The optical illusion in the picture which we reproduce is due to the defective drawing of the two men on the platform. In actual size upon the paper the further man looks much taller than the other.

Illusion

Measurement, however, shows the figures to be exactly of a height. This illusion is due to the fact that the head of the further man is quite out of perspective. If he is about as tall as the other, and on level ground, both heads should be about on the same line. As drawn, he is, in fact, a monster more than eight feet high.

22. If Dick, who is five feet in height, stands bolt-upright in a swing, the ropes of which are twenty feet long, how much further in round numbers do his feet travel than his head in describing a semi-circle?

Solution

Here is an excellent and very simple illustration of a well-known opticalcuriosity:—

Illusion

Hold this picture at arm’s length in the right hand, hold the left hand over the left eye, and draw the picture towards you gradually, looking always at the black cross with the right eye. The black disc will presently disappear, and then come into sight again as you continue to advance the paper.

23. Can you name nine countries in Europe of which the initial letters are the same as the finals?

Solution

Here is a delightfully simple way in which market gardeners, or others who buy or sell weighty produce, can check their invoices for potatoes or what not.

Say, for example, that a consignment weighs 6 tons, 10 cwts., 1 qr. Then, since 20 cwts. are to a ton as 20s. are to a pound, and each quarter would answer on these lines to 3d., we can at once write down £6 10s. 3d., as the price at £1 the ton. On this sure basis any further calculation is easily made.

SkullHOW TO SEE THE GHOST

Skull

HOW TO SEE THE GHOST

Look steadily, in a good light, for thirty seconds at the cross in the eye of the pictured skull; then look up at the wall or ceiling, or look fixedly at a sheet of paper for another thirty seconds, when a ghost-like image of the skull will be developed.

A gardener, when he had planted 100 trees on a line at intervals of 10 yards, was able to walk from the first of these to the last in a few seconds, for they were seton the circumference of a circle!

Here is another example of what is known as the persistence ofvision:—

Illusion

Look fixedly for some little time at this grotesque figure, then turn your eyes to the wall or ceiling, and you will in a few seconds see it appear in dark form upon a light ground.

The sum of nine figures a number will make,Of which just a third will remainIf fifty away from the whole you should take,Thus turning a loss to a gain.

It needs something more than mere arithmetic to discover that the solution to this puzzle is XLV, the sum of the nine digits, for if the L is removed, XV, the third of XLV, remains.

Here is another curiousillusion:—

Illusion

The four straight lines are perfectly parallel, but the contradictory herring-bones disturb the eye.

If our penny had been current coin in the first year of the Christian era, and had been invested at compound interest at five per cent., it would have amounted in 1905 to more than £132,010,000,000,000,000,000,000,000,000,000,000,000.

This gigantic sum would afford an income of £101,890,000,000,000,000,000 every second to every man, woman, and child in the world, if we take its population to be 1,483,000,000 souls!

Absurdly small in contrast to these startling figures is the modest eight shillings which the same penny would have yielded in the same time at simple interest.

Here in another form is shown the illusion of length.

Illusion

At first sight it seems that the two upright lines are distinctly longer than the line that slopes, but it is not so.

Here is a neat method of discovering the age of a person older thanyourself:—

Subtract your own age from 99. Ask your friend to add this remainder to his age, and then to remove the first figure and add it to the last, telling you the result. This will always be the difference of your ages. Thus, if you are 22, and he is 35, 99 - 22 = 77. Then 35 + 77 = 112. The next process turns this into 13, which, added to your age, gives his age, 35.

In this diagram one hundred and twenty-one circular spots are grouped in a diamond.

Illusion

If we half close our eyes, and look at this through our eye-lashes, we find that it takes on the appearance of a section of honeycomb, with hexagonal cells.

Here is a ready method for multiplying together any two numbers between 12 and 20.

Take one of the two numbers and add it to the unit digit of the other. Beneath the sum thus obtained, but one place to the right, put the product of the unit digits of the two original numbers.

The sum of these new numbers is the product of the numbers that were chosen.Thus:—

BLACKChessWHITE

BLACK

Chess

WHITE

Black has made an illegal move. He must replace this, and move his king as the penalty. White then mates on the move.

Solution

Here is a method for determining the total number of balls in a solid pyramid built up on a squarebase:—

Multiply the number of shot on one side of the base line by 2, add 3, multiply by the number on the base line, add 1, multiply again by the number on the base, and finally divide by 6. Thus, if the base line is12—

12 × 2 + 3 = 27; 27 × 12 + 1 = 325; 325 × 12 = 3900; and 3900 ÷ 6 = 650, which is the required number.

[1]N.B.—This title does not imply a tragedy.

BLACKChessWHITE

BLACK

Chess

WHITE

White might have given mate on the last move. White now to retract his move, and mate at once.

Show by analysis the mating position.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play and draw.

Solution

“Now, boys,” said Dr Bulbous Roots to class, “you shall have a half-holiday if you prove in a novel way that 10 is an even number.”

Next morning, when the doctor came into school, he found this on theblackboard:—

Q. E. D.(Quite easily done!)

The half-holiday was won.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and draw.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

There are twelve variations in this beautiful problem.

Solution

It will interest all who study short cuts and contrivances to know that a novice at arithmetic who has mastered simple addition, and can multiply or divide by 2, but by no higher numbers, can, by using all these methods, multiply any two numbers together easily and accurately.

This is how it isdone:—

Write down the numbers, say 53 and 21, divide one of them by 2 as often as possible, omitting remainders, and multiply the other by 2 the same number of times; set these down side by side, as in the instance given below, and wherever there is anevennumber on the division side, strike out the corresponding number on the multiplication side. Add up what remains on that side, and the sum is done.Thus:—

which is 53 multiplied by 21.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

Solution

The following particulars about a very rare property of numbers will be new and interesting to many of ourreaders:—

The number 6 can only be divided without remainder by 1, 2, and 3, excluding 6 itself. The sum of 1 + 2 + 3 is 6. The only exact divisors of 28 are 1, 2, 4, 7, and 14, and the sum of these is 28; 6 and 28 are therefore known as perfect numbers.

The only other known numbers which fulfil these conditions are 496; 8128; 33,550,336; 8,589,869,056; 137,438,691,328; and2,305,843,008,139,952,128.This most remarkable rarity of perfect numbers is a symbol of their perfection.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves. There are no less than twelve variations!

Solution

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

Solution

Somewhat akin to perfect numbers are what are known as amicable numbers, of which there is a still smaller quantity in the realm of numbers.

The number 220 can be divided without remainder only by 1, 2, 4, 5, 10, 11, 22, 44, 55, and 110, and the sum of these divisors is 284. The only divisors of 284 are 1, 2, 4, 71, and 142, and the sum of these is 220.

The only other pairs of numbers which fulfil this curious mutual condition, that the sum of the divisors of each number exactly equals the other number, are 17,296 with 18,416, and 9,363,584 with 9,437,056. No other numbers, at least below ten millions, are in this way “amicable.”

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in two moves.

Solution

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

There is a pleasant touch of mystery in the following method of discovering a person’s age:—Ask any such subjects of your curiosity to write down the tens digit of the year of their birth, to multiply this by 5, to add 2 to the product, to multiply this result by 2, and finally to add the units digit of their birth year. Then, taking the paper from them, subtract the sum from 100. This will give you their age in 1896, from which their present age is easily determined.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

As I came in after a day among the birds and rabbits, the keeper asked me—“Well, sir, what sport?” I replied, “36 heads and 100 feet.” It took him some time to calculate that I had accounted for 22 birds and 14 rabbits.

BLACKChessWHITE

BLACK

Chess

WHITE

White to play, and mate in three moves.

Solution

The nine digits can be arranged to form fractions equivalent to

13141516171819

thus:—

582317469=13795631824=14297314865=15294317658=16527436918=17932174568=18836175249=19

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

This motor problem will be new and amusing to manyreaders:—

Letmbe the driver of a motor-car, working with velocityv. If a sufficiently high value is given tov, it will ultimately reachpc. In most casesvwill then =o. For low values ofv,pcmay be neglected; but ifvbe large it will generally be necessary to squarepc, after whichvwill again assume a positive value.

By a well-known elementary theorem,pc+lsd= (pc)2, but the squaring may sometimes be effected by substitutingx3(or × × ×) forlsd. This is preferable, iflsdis small with regard tom. Iflsdbe made sufficiently large,pcwill vanish.

Now ifjpbe substituted forpc(which may happen if the difference betweenmandpcbe large) the solution of the problem is more difficult. No value oflsdcan be found to effect the squaring ofjp, for, as is well-known, (jp)2is an impossible quantity.

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

To divide any sum easily by 99, cut off the two right-hand figures of the dividend and add them to all the others. Set down the result of this in line below, and then repeat this process until no figures remain on the left to be thus dealt with.

Now draw a line down between the tens and hundreds columns, and add all up on the left of it,thus:—

The last number on the right of the lines shows always the remainder. If this should appear as 99 (as in the second example above), add one to the number on the left.

Back to Index Next