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If the number 37 be multiplied by 3, or any multiple of 3 up to 27, the product is expressed by three similar digits. Thus—

 37 × 3 = 111

 37 × 6 = 222

 37 × 9 = 333

The products succeed each other in the order of the digits read downwards, 1, 2, 3, etc., these being multiplied by 3 (their number of places) reproduce the multiplicand of 37.

 1 × 3 = 3

 2 × 3 = 6

 3 × 3 = 9

If it be multiplied by multiples of 3, beyond 27, this peculiarity is continued, except that the extreme figures taken together represent the multiple of 3 that is used as a multiplier. Thus—

 37 × 30 = 1110, extreme figures, 10

 37 × 33 = 1221 " " 11

 37 × 36 = 1332 " " 12

The number 73 (which is 37 inverted) multiplied by each of the numbers of arithmetical progression 3, 6, 9, 12, 15, etc., produces products terminating (unit’s place) by one of the ten different figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These figures will be found in the reverse order to that of the progression, 73 × 3 produces 9, by 6 produces 8, and 9 produces 7, and so on.

Another number which falls under some mysterious law of series is 142,857, which, multiplied by 1, 2, 3, 4, 5, or 6 gives the same figures in the same order, beginning differently; but if multiplied by 7, gives all 9’s.

142,857	multiplied by	1	= 142,857
"	"	2	= 285,714
"	"	3	= 428,571
"	"	4	= 571,428
"	"	5	= 714,285
"	"	6	= 857,142
"	"	7	= 999,999

Multiplied by 8, it gives 1,142,856, the first figure added to the last makes the original number—142,857.

The vulgar fraction ¹⁄₇ = ·142,857.

The following number, 526315789473684210, if multiplied as above, will, in the product, present the same peculiarities, as also will the number 3448275862068965517241379310.

The multiplication of	987654321	by 45	= 444444444445
Do.	123456789	" 45	= 5555555505
Do.	987654321	" 54	= 53333333334
Do.	123456789	" 54	= 6666666606

Taking the same multiplicand and multiplying by 27 (half 54) the product is 26,666,666,667, all 6’s except the extremes, which read the original multiplier (27). If 72 be used as a multiplier, a similar series of progression is produced.

6. In stables five, can you contrive to put in horses twenty— In each stable an odd horse, and not a stable empty?

“THREE THREES ARE TEN.”

This little trick often puzzles many:—

Place three matches, coins, or other articles on the table, and by picking each one up and placing it back three times, counting each time to finish with number 10, instead of 9. Pick up the first match and return it to the table saying 1; the same with the second and third, saying 2 and 3; repeat this counting 4; but the fifth match must be held in the hand, saying at the time it is picked up, 5; the other two are also picked up and held in hand, making 6 and 7; the three matches are then returned to the table as 8, 9, and 10. If done quickly few are able to see through it.

7. A man bought a colt for a certain sum and sold him 2 years afterwards for £50 14s., gaining thereby as much per cent. per annum compound interest as it had cost him. What was the original price?

Do Figures Lie?

“Figures cannot lie,” is a very old saying. Nevertheless, we can all be deceived by them. Perhaps one of the best instances of them leading us astray is the following:—

An employer engaged two young men, A and B, and agreed to pay them wages at the rate of £100 per annum. A enquires if there is to be a “rise,” and is answered by the employer, “Yes, I will increase your wages £5 every six months.” “Oh! that is very small; it’s only £10 per year,” replied A. “Well,” said the employer, “I will double it, and give you a rise of £20 per year.” A accepts the situation on those terms.

B, in making his choice, prefers the £5 every six months. At the first glance, it would appear that A’s position was the better.

Now, let us see how much each receives up to the end of four years:—

A	B
1st year	£100	50}	1st year
2nd "	120	55}
3rd "	140	60}	2nd "
4th "	160	65}
		70}	3rd "
		75}
		80}	4th "
		85}
	£520	£540

A spieler at a Country Show amused the people with the following game:—He had 6 large dice, each of which was marked only on one face—the first with 1, the second 2, and so on to the sixth, which was marked 6. He held in his hand a bundle of notes, and offered to stake £100 to £1 if, in throwing these six dice, the six marked faces should come up only once, and the person attempting it to have 20 throws.

Though the proposal of the spieler does not on the first view appear very disadvantageous to those who wagered with him, it is certain there were a great many chances against them.

The six dice can come up 46,656 different ways, only one of which would give the marked faces; the odds, therefore, in doing this in one throw would be 46,655 to 1 against, but, as the player was allowed 20 throws, the probability of his succeeding would be—

20

46,656

To play an equal game, therefore, the spieler should have engaged to return 2332 times the money deposited.

TREBLE RULE OF THREE.

If 70 dogs with 5 legs each catch 90 rabbits with 3 legs each in 25 minutes, how many legs must 80 rabbits have to get away from 50 dogs with 2 legs each in half an hour?

8. Suppose a greyhound makes 27 springs whilst a hare makes 25, and the springs are equal: if the hare is 50 springs before the hound at the start, in how many springs will the hound overtake the hare?

The first Arithmetic in English was written by Tonstal, Bishop of London, and printed by Pinson in 1552.

Two persons playing dominoes 10 hours a day and making 4 moves a minute could continue 118,000 years without exhausting all the combinations of the game.

A schoolmaster wrote the word “dozen” on the blackboard, and asked the pupils to each write a sentence containing the word. He was somewhat taken aback to find on one of the slates the following unique sentence: “I dozen know my lesson.”

9. I have a piece of ground, which is neither square nor round, But an octagon, and this I have laid out In a novel way, though plain in appearance, and retain Three posts in each compartment; but I doubt Whether you discover how I apportioned it, e’en tho’ I inform you ’tis divided into four. But, if you solve it right, ’twill afford you much delight And repay you for the trouble, I am sure.

At an examination in arithmetic, a little boy was asked “what two and two made?” Answer—“Four.” “Two and four?” Answer—“Six.” “Two and six?” Answer—“Half-a-crown.”

10. A certain gentleman dying left his executor the sum of £3,000 to be disposed of in the following manner, viz.:—To give to his son £1,000, to his wife £1,000, to his sister £1,000, and to his sister’s son £1,000, to his mother’s grandson £1,000, to his own father and mother £1,000, and to his wife’s own father and mother £1,000—required, the scheme of kindred.

COPY OF LETTER FROM FIRM TO COMMERCIAL TRAVELLER.

Sydney,

25th Jan., 1895.

Mr. Einstein, Townsville,

Dear Sir,

Ve hav receved your letter on the 18th mit expense agount and round list. vat ve vants is orders, ve haf plenty maps in Sydney vrom vich to make up round lists also big families to make expenses.

Mr. Einstein ve find in going through your expenses agount 10s. for pilliards please don’t buy no more pilliards for us. vat ve vants is orders, also ve do see 30s. for a Horse and Buggy, vere is de horse and vot haf you done mit de Buggy the rest on your expenses agount vas nix but drinks—vy don’t you suck ice. ve sended you to day two boxes cigars, 1 costed 6/- and the oder 3/6 you can smoke the 6/- box, but gif de oders to your gustomers, ve send you also samples of a necktie vat costed us 28/- gross, sell dem for 30/- dozen if you can’t get 30/- take 8/6, vat ve vants is orders. The neckties is a novelty as ve hav dem in stock for seven years and ain’d sold none. My brother Louis says you should stop in Rockhampton. His cousin Marks livs dere. Louis says you should sell Marks a good bill; dry him mit de neckties first, and sell mostly for cash, he is Louis’s cousin. Ve only giv credit to dem gustomers vat pays cash. Don’t date any more bills ahead, as the days are longer in the summer as in the vinter. Don’t show Marks any of the good sellers, and finaly remember Mr. Einstein mit us veder you do bisness or you do nothings at all vat ve vants is orders.

Yours Truly,

Shadrack & Co.

P.S.—Keep the expenses down.

11. Two fathers and two sons went into a hotel to have drinks, which amounted to one shilling. They each spent the same amount. How much did each pay?

12. In a cricket match, a side of 11 men made a certain number of runs. One obtained one-eighth of the number, each of two others one-tenth, and each of three others one-twentieth. The rest made up among them 126 (the remainder of the score), and four of the last scored five times as many as the others. What was the whole number of runs, and the score of each man?

BRAINS v. BRAWN.

Schoolmaster—“What is meant by mental occupation?”

Pupil—“One in which we use our minds.”

Schoolmaster—“And a manual occupation?”

Pupil—“One in which we use our hands.”

Schoolmaster—“Now, which of these occupations is mine. Come, now; what do I use most in teaching you?”

Pupil (quickly)—“Your cane, sir!”

MAGIC ADDITION.

To write the answer of an addition sum, when only one line has been written.

Tell a person to write down a row of figures. Now, this row will constitute the main body of the answer. Tell him to write another row beneath it; you now write a row also, matching his second row in pairs of 9’s he writes one more row, and you again supply another in the same manner. Your addition sum will now consist of five lines, four of which are paired; the first line, or key line, being the answer to the sum.

From the unit figure in the key line deduct the number of pairs of 9’s—in this instance two—and place the remainder, 6, as the unit figure of the answer, then write in order the rest of the figures in the key line, annexing the 2 to the extreme left; this will constitute the complete answer.

It, of course, is not necessary to adhere to two pairs of 9’s; there may be three, four, or even more; but the total number of lines, including the key line, must be odd, and the number of pairs must be deducted from the unit figure of the key line, and this same number be written down at the extreme left. The number of figures in each line should always be the same. As the location of the key line may be changed if necessary, the artifice could not easily be detected.

Punctuation was first used in literature in the year 1520. Before that time wordsandsentenceswereputtogetherlikethis.

13. Smith and Brown meet a dairymaid with a pail containing milk. Smith maintains that it is exactly half full; Brown that it is not. The result is a wager. They have no instrument of any kind, nor can they procure one by means of which to decide the wager; nevertheless they manage to find out accurately, and without assistance, whether the pail is half-full or not. How is it done?—It should be added that the pail is true in every direction.

A HINT FOR TAILORS.

“There, stand in that position, please, and look straight at that notice while I take your measure.”

Customer reads the notice—

“Terms Cash.”

NUMBER 9.

If two numbers divisible by 9 be added together the sum of the figures in the amount will be either 9 or a number divisible by 9.

Example:	54
(1)	36
	90

If one number divisible by 9 be subtracted from another number divisible by 9, the remainder will be either a 9 or a number divisible by 9.

Example:	72
(2)	18
	54

If one number divisible by 9 be multiplied by another number divisible by 9, the product will be divisible by 9.

Example:	54
(3)	27
	1458

If one number divisible by 9 be divided by another number divisible by 9, the quotient will be divisible by 9.

Example:	27	) 3645
(4)
		135

In the above examples it is worth noting that the figures in each answer added together continually produce 9.

(1) 90 = 9 (2) 54 = 9 (3) 1458 = 18 = 9 (4) 135 = 9

Also, if these answers be multiplied by any number whatever, a similar result will be produced.

Example: 135 x 8 = 1080 = 9

If any row of two or more figures be reversed and subtracted from itself, the figures composing the remainder will, when added, be a multiple of 9, and if added together continually will result in 9.

Example:	7362
	2637
	4725	= 18 = 9

Tell a person to write a row of figures, then to add them together, and to subtract the total from the row first written, then to cross out any one of the figures in the answer, and to add the remaining figures in the answer together, omitting the figure crossed out; if the total be now told, it is easy to discover the figure crossed out.

Example:	4367256	= 33
	33
	4367223	= 27

It should be observed that the figures of the answer to the subtraction when added together equal 27—a multiple of 9; this, of course, is always the case. Now, suppose that 7 was the figure crossed out, then the sum of the figures in the answer (omitting 7) would be 20; this number being told by the person, it is easily seen that 7 must have been crossed out, as that figure is required to complete the multiple 27. If after the figure has been crossed out, the remaining figures total a multiple of 9, it is evident that either a cipher or a 9 must have been the figure erased.

Multiply the digits—omitting 8—by any multiple of 9, and the product will consist of that multiple,

Example:	12345679	36= 4 x 9
	36
	444444444

If a figure with a number of ciphers attached to it be divided by 9, the quotient will be composed of that figure only repeated as many times as there are ciphers in the dividend; with the same figure as the remainder.

Example:	9	) 7000000
		————
		777777 - 7

EXCUSES.

“Miss Brown,—You must stop teach my Lizzie fisical torture. She needs reading and figgers more an that. If I want her to do jumpin I kin make her jump.”

“Please let Willie home at 3 o’clock. I take him out for a little pleasure, to see his father’s grave.”

“Dear Teecher,—Please excuse John for staying home—he had the meesels to oblige his father.”

“Dear Miss——, Please excuse my boy scratching hisself, he’s got a new flannel shirt on.”

“A country schoolmaster received from a small boy a slip of paper which was supposed to contain an excuse for the non-attendance of the boy’s brother. He examined the paper, and saw thereon:

“Kepatomtogoataturing.”

Unable to understand, the small boy explained to the master that his big brother had been “kept at home to go taturing”—that is, to dig potatoes.

“Tommy,” said the school teacher, “you must get your father to give you an excuse the next time you stay away from school.”

“That’s no use, teacher. Dad’s no good at making excuses; mother bowls him out every time.”

HARVESTING.

14. A and B engage to reap a field for 90s. A could reap it in 9 days by himself; they promised to complete it in five days; they found, however, that they were obliged to call in C (an inferior workman) to assist them the last two days, in consequence of which B received 3s. 9d. less than he otherwise would have done. In what time could B and C reap the field alone?

15. A man has a triangular block of land, the largest side being 136 chains, and each of the other sides 68 chains. What is the value of the grass on it, at the rate of £2 an acre?

A school inspector in the North of Ireland was once examining a geography class, and asked the question:

“What is a lake?”

He was much amused when a little fellow, evidently a true gem of the emerald isle, answered: “It’s a hole in a can, sur.”