Freaks of Figures

A Handy Short Cut

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.

No. XXI.—The Persistence of Vision

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.

Not So Fast!

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 set on the circumference of a circle!

No. XXII.—The Persistence of Vision

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

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.

A Puzzle Number

The sum of nine figures a number will make,
Of which just a third will remain
If 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.

No. XXIII.—Another Parallel Freak

Here is another curious illusion:—

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

The Magic of Figures

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.

No. XXIV.—Illusion of Length

Here in another form is shown the illusion of length.

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

What Is Your Age?

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

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.

No. XXV.—The Honeycomb Illusion

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

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.

Multiplication No Vexation

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:—

19 × 13. (19 + 3) = 22

(9 × 3) = 27

247

No. XXVI.—CHESS CAMEO By Dr Gold A Chess Joke

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

Solution

Shot in a Pyramid

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

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 is 12—

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

No. XXVII.—CHESS CAMEO By A. Cyril Pearson A Chess Puzzle

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

No. XXVIII.—CHESS CAMEO By J. G. Campbell A very Clever Device

White to play and draw.

Solution

Comic Arithmetic

“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 the blackboard:—

SIX = 6

SIX = 9

By subtraction S = -3

SEVEN = 7

S = -3

Therefore SEVEN = 10

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

The half-holiday was won.

No. XXIX.—CHESS CAMEO By E. B. Cook A Fine Example

White to play, and draw.

Solution

No. XXX.—CHESS CAMEO By A. F. Mackenzie A Prize Problem

White to play, and mate in two moves.

There are twelve variations in this beautiful problem.

Solution

Magic Multiplication

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 is done:—

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 an even number 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:—

53 21)

26 (42)

13 84)

6 (168)

3 336)

1 672)

1113)

which is 53 multiplied by 21.

No. XXXI.—CHESS CAMEO By A. W. Galitzky A Prize Problem

White to play, and mate in two moves.

Solution

No. XXXII.—Chess Cameo By S. Loyd

White to play, and mate in two moves.

Solution

No. XXXIII.—CHESS CAMEO By B. G. Laws A Prize Problem

White to play, and mate in two moves.

Solution

Perfect Numbers

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

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; and 2,305,843,008,139,952,128. This most remarkable rarity of perfect numbers is a symbol of their perfection.

No. XXXIV.—Chess Cameo By Emil Hoffmann

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

Solution

No. XXXV.—CHESS CAMEO By J. Pospicil A Prize Problem

White to play, and mate in two moves.

Solution

Amicable Numbers

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.”

No. XXXVI.—CHESS CAMEO By H. J. C. Andrews A Prize Problem

White to play, and mate in two moves.

Solution

No. XXXVII.—CHESS CAMEO By Alfred de Musset A Gem of the First Water

White to play, and mate in three moves.

Solution

A Swarm of Ones

1 × 9 + 2 = 11

12 × 9 + 3 = 111

123 × 9 + 4 = 1111

1234 × 9 + 5 = 11111

12345 × 9 + 6 = 111111

123456 × 9 + 7 = 1111111

1234567 × 9 + 8 = 11111111

12345678 × 9 + 9 = 111111111

No. XXXVIII.—CHESS CAMEO By Frank Healey A Masterpiece

White to play, and mate in three moves.

Solution

Divination by Figures

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.

No. XXXIX.—CHESS CAMEO By Frank Healey The “Bristol Prize Problem”

White to play, and mate in three moves.

Solution

Fur and Feathers

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.

No. XL.—CHESS CAMEO By J. E. Campbell Splendid Strategy

White to play, and mate in three moves.

Solution

Juggling with the Digits

The nine digits can be arranged to form fractions equivalent to

13 14 15 16 17 18 19

thus:—

582317469 = 13 795631824 = 14 297314865 = 15 294317658 = 16 527436918 = 17 932174568 = 18 836175249 = 19

No. XLI.—Chess Cameo By W. Grimshaw

White to play, and mate in three moves.

Solution

No. XLII.—Chess Cameo By S. Loyd

White to play, and mate in three moves.

Solution

A Motor Problem

This motor problem will be new and amusing to many readers:—

Let m be the driver of a motor-car, working with velocity v. If a sufficiently high value is given to v, it will ultimately reach pc. In most cases v will then = o. For low values of v, pc may be neglected; but if v be large it will generally be necessary to square pc, after which v will again assume a positive value.

By a well-known elementary theorem, pc + lsd = (pc), but the squaring may sometimes be effected by substituting x (or × × ×) for lsd. This is preferable, if lsd is small with regard to m. If lsd be made sufficiently large, pc will vanish.

Now if jp be substituted for pc (which may happen if the difference between m and pc be large) the solution of the problem is more difficult. No value of lsd can be found to effect the squaring of jp, for, as is well-known, (jp) is an impossible quantity.

No. XLIII.—Chess Cameo By J. G. Campbell

White to play, and mate in three moves.

Solution

No. XLIV.—Chess Cameo By Frank Healey

White to play, and mate in three moves.

Solution

A Neat Method of Division

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:—

8694 32 120 78

87 26 1 98

1 13 99

14 121 and 99 over.

8782 and 14 over. In other words, 122.

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.

No. XLV.—Chess Cameo By Blumenthal and Kund

White to play, and mate in three moves.

Solution

No. XLVI.—CHESS CAMEO By A. F. Mackenzie A Prize Problem

White to play, and mate in three moves.

Solution

A Smart Schoolboy

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

1,000,000 1,000,000 1,000,000

19 19 19

999,981 999,981 999,981

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.

No. XLVII.—Chess Cameo By A. Cyril Pearson

White to play, and mate in three moves.

Solution

No. XLVIII.—CHESS CAMEO By Frank Healey Quite a Gem

White to play, and mate in three moves.

Solution

Lewis Carroll’s Short Cut

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 = 4051,0 36459 = 4051.

No. XLIX.—Chess Cameo By A. Bayersdorfer

White to play, and mate in three moves.

Solution

Another Freak of Figures

1 × 8 + 1 = 9

12 × 8 + 2 = 98

123 × 8 + 3 = 987

1234 × 8 + 4 = 9876

12345 × 8 + 5 = 98765

123456 × 8 + 6 = 987654

1234567 × 8 + 7 = 9876543

12345678 × 8 + 8 = 98765432

123456789 × 8 + 9 = 987654321

No. L.—Chess Cameo By J. Dobrusky

White to play, and mate in three moves.

Solution

Divination by Numbers

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.

No. Li.—Chess Cameo By Konrad Bayer

White to play, and mate in three moves.

Solution

Coincidences

Here is a curious rough rule for remembering distances and sizes:—

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.

No. LII.—Chess Cameo By J. Berger

White to play, and mate in three moves.

Solution

Personal Arithmetic

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!”

No. LIII.—Chess Cameo By H. F. L. Meyer

White to play, and mate in three moves.

Solution

Division by Subtraction

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:—

363 56408 375034

0 0 0

33 5128 34094

No. LIV.—Chess Cameo By Frank Healey

White to play, and mate in three moves.

Solution

Luck in Odd Numbers

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:—

1 + 3 + 5 = 9 = 3 × 3.

1 + 3 + 5, etc., up to 17 = 81 = 9 × 9.

1 + 3 + 5, etc., up to 99 = 2500 = 50 × 50.

No. Lv.—Chess Cameo By Frank Healey

White to play, and mate in three moves.

Solution

The Versatile Number

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:—

285714 = 142857 × 2

428571 = 142857 × 3

571428 = 142857 × 4

714285 = 142857 × 5

857142 = 142857 × 6

No. LVI.—Chess Cameo By J. E. Campbell

White to play, and mate in three moves.

Solution

A Paradox

By the following simple method, a plausible attempt is made to prove that 1 is equal to 2:—

Suppose that a = b, then

ab = a

∴ ab - b = a - b

∴ b(a - b) = (a + b)(a - b)

∴ b = a + b

∴ b = 2b

∴ 1 = 2

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

No. LVII.—CHESS CAMEO Double First Prize By A. Cyril Pearson

White to play, and mate in four moves.

Solution

Quick Calculation

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 a day:—

£7 0 0

3 10 0

2 11

£10 12 11

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.