Volume 1 — VAGARIES OF MATHEMATICS.

"As dull as arithmetic" is a phrase that is familiar to almost every schoolboy, and is a figure of comparison that is frequently evoked by those sages who hold down empty cracker-boxes in rural general stores. The fact is, however, that arithmetic is not always half so dull as it looks. Like some of those persons who earn a livelihood by teaching it to the young, it has a dry humor and a few vagaries of its own.

One of these vagaries has to do with the figure 9, and it is thus described by William Walsh in his "Handy Book of Literary Curiosities":

It is a most romantic number, and a most persistent, self-willed, and obstinate one. You cannot multiply it away or get rid of it anyhow. Whatever you do, it is sure to turn up again, as did the body of Eugene Aram's victim.

A mathematician named Green, who died in 1794, is said to have first called attention to the fact that all through the multiplication table the product of nine comes to nine. Multiply by any figure you like, and the sum of the resultant digits will invariably add up as nine. Thus, twice 9 is 18; add the digits together, and 1 and 8 make 9. Three times 9 is 27; and 2 and 7 is 9. So it goes on up to 11 times 9, which gives 99. Very good. Add the digits together; 9 and 9 is 18, and 8 and 1 is 9.

Go on to any extent, and you will find it impossible to get away from the figure 9. Take an example at random: 9 times 339 is 3,051; add the digits together, and they make 9. Or again, 9 times 2,127 is 19,143; add the digits together, they make 18, and 8 and 1 is 9. Or still again, 9 times 5,071 is 45,639; the sum of these digits is 27, and 2 and 7 is 9.

This seems startling enough. Yet there are other queer examples of the same form of persistence. It was M. de Maivan who discovered that if you take any row of figures, and, reversing their order, make a subtraction sum of obverse and reverse, the final result of adding up the digits of the answer will always be 9 As, for example:

2941

Reverse,1492

——

1449

Now. 1 + 4 + 4 + 9 = 18; and 1 + 8 = 9.

The same result is obtained if you raise the numbers so changed to their squares or cubes. Start anew, for example, with 62; reversing it, you get 26. Now, 62 - 26 = 36, and 3 + 6 = 9. The squares of 26 and 62 are, respectively, 676 and 3844. Subtract one from the other, and you get 3168 = 18, and 1 + 8 = 9.

So with the cubes of 26 and 62, which are 17,576 and 238,328. Subtracting, the result is 220,752 = 18, and 1 + 8 = 9.

Again, you are confronted with the same puzzling peculiarity in another form. Write down any number, as, for example, 7,549,132, subtract therefrom the sum of its digits, and, no matter what figures you start with, the digits of the product will always come to 9.

7549132, sum of digits = 31. 31 ——— 7549101, sum of digits = 27, and 2 + 7 = 9.

Again, set the figure 9 down in multiplication, thus:

1 × 9 = 9 2 × 9 = 18 3 × 9 = 27 4 × 9 = 36 5 × 9 = 45 6 × 9 = 54 7 × 9 = 63 8 × 9 = 72 9 × 9 = 81 10 × 9 = 90

Now, you will see that the tens column reads down 1, 2, 3, 4, 5, 6, 7, 8, 9, and the units column up 1, 2, 3, 4, 5, 6, 7, 8, 9.

Here is a different property of the same number. If you arrange in a row the cardinal numbers from 1 to 9, with the single omission of 8, and multiply the sum so represented by any one of the figures multiplied by 9, the result will present a succession of figures identical with that which was multiplied by 9. Thus, if you wish a series of fives, you take 5 × 9 = 45 for a multiplier, with this result:

12345679 45 ———— 61728395 49382716 ———— 555555555

A very curious number is 142,857, which, multiplied by 1, 2, 3, 4, 5, or 6, gives the same figures in the same order, beginning at a different point, but if multiplied by 7 gives all nines. Multiplied by 1, it equals 142,857; multiplied by 2, equals 285,714; multiplied by 3, equals 428,571; multiplied by 4, equals 571,428; multiplied by 5, equals 714,285; multiplied by 6, equals 857,142; multiplied by 7, equals 999,999.

Multiply 142,857 by 8, and you have 1,142,856. Then add the first figure to the last, and you have 142,857, the original number, the figures exactly the same as at the start.

The number 37 has this strange peculiarity: multiplied by 3, or by any multiple of 3 up to 27, it gives three figures all alike. Thus, three times 37 will be 111. Twice three times (6 times) 37 will be 222; three times three times (9 times) 37 gives three threes; four times three times (12 times) 37, three fours, and so on.

The wonderfully procreative power of figures, or, rather, their accumulative growth, has been exemplified in that familiar story of the farmer, who, undertaking to pay his farrier one grain of wheat for the first nail, two for the second, four for the third, and so on, found that he had bargained to give the farrier more wheat than was grown in all England.

My beloved young friends who love to frequent the roulette-table, do you know that if you begin with a dime, and were allowed to leave all your winnings on the table, five consecutive lucky guesses would give you almost a million and a half of dollars, or, to be exact, $1,450,625.52?

Yet that would be the result of winning thirty-five for one five times hand-running.

Here is another example. Take the number 15, let us say. Multiply that by itself, and you get 225. Now multiply 225 by itself, and so on until fifteen products have been multiplied by themselves in turn.

You don't think that is a difficult problem? Well, you may be a clever mathematician, but it would take you about a quarter of a century to work out this simple little sum.

The final product called for contains 38,589 figures, the first of which are 1,442. Allowing three figures to an inch, the answer would be more than a thousand feet long. To perform the operation would require about 500,000,000 figures. If they can be made at the rate of one a minute, a person working ten hours a day for three hundred days in each year would be twenty-eight years about it.

NUMBERS THAT EQUIVOCATED.

The Woman's Home Companion repeats a good story that is told of a quick-witted Irishman with a natural aptitude for mental arithmetic who was working in a field with a Dutchman, when they unearthed a box of silver coins. The covetous Dutchman at once laid claim to the whole booty, because he was the first to break it open and discover its valuable contents.

"Go softly," said the Irishman, "for the whole business is mine. It's a bit of money that was left me by an uncle, and I buried it here for safe-keeping. There was a thousand dollars."

"All right with that," replied the Dutchman, as he caught on to the bait. "If you tell me how much money there is, it's with you; if you miss, she's mine."

"That's fair, and you have the sentiments of a gentleman," replied Pat, as he made a quick mental calculation from the weight of the box that there must be somewhere between fifty and three hundred and fifty dollars. "I sent six hundred and forty-two dollars and fifty-three cents to my mother in the old country, so add that amount to what there is in the box."

"That is done so quick," said the Dutchman.

"Then deduct that amount from the sum of one thousand dollars, which was left me," said Pat.

"Done again," said the Dutchman.

"Now deduct those figures from three hundred and fifty-seven dollars and forty-seven cents, which I had to pay the lawyers, and it leaves the exact amount to one cent that you will find in the box."

"That's the right money to a penny," said the Dutchman, after he had counted it carefully, "and it proves that you are an honest man."

It is not every one who can see through the mathematics of this puzzle so as to know that Pat's problem would work out all right if the box contained any sum up to three hundred and fifty dollars.

FREAK COMBINATIONS.

A well-known professor has drawn attention to the following series of numbers, which are here given without remark:

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 123456789 × 9 + 9 = 111111111 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