Part 4

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.