# Amusements in Mathematics

"Nine worthies were they called."

DRYDEN: The Flower and the Leaf.

I give these puzzles, dealing with the nine digits, a class to themselves, because I have

always thought that they deserve more consideration than they usually receive. Beyond

the mere trick of "casting out nines," very little seems to be generally known of the laws

involved in these problems, and yet an acquaintance with the properties of the digits often

supplies, among other uses, a certain number of arithmetical checks that are of real value

in the saving of labour. Let me give just one example—the first that occurs to me.

If the reader were required to determine whether or not 15,763,530,163,289 is a square

number, how would he proceed? If the number had ended with a 2, 3, 7, or 8 in the digits

place, of course he would know that it could not be a square, but there is nothing in its

apparent form to prevent its being one. I suspect that in such a case he would set to work,

with a sigh or a groan, at the laborious task of extracting the square root. Yet if he had

given a little attention to the study of the digital properties of numbers, he would settle

the question in this simple way. The sum of the digits is 59, the sum of which is 14, the

sum of which is 5 (which I call the "digital root"), and therefore I know that the number

cannot be a square, and for this reason. The digital root of successive square numbers

from 1 upwards is always 1, 4, 7, or 9, and can never be anything else. In fact, the series,

1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The analogous series for triangular

numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. So here we have a similar negative check, for a number

cannot be triangular (that is, (n²+n)/2) if its digital root be 2, 4, 5, 7, or 8.

76.—THE BARREL OF BEER.

A man bought an odd lot of wine in barrels and one barrel containing beer. These are

shown in the illustration, marked with the number of gallons that each barrel contained.

He sold a quantity of the wine to one man and twice the quantity to another, but kept the

beer to himself. The puzzle is to point out which barrel contains beer. Can you say which

one it is? Of course, the man sold the barrels just as he bought them, without

manipulating in any way the contents.

77.—DIGITS AND SQUARES.

It will be seen in the diagram that we have so arranged the nine digits in a square that the

number in the second row is twice that in the first row, and the number in the bottom row

three times that in the top row. There are three other ways of arranging the digits so as to

produce the same result. Can you find them?

78.—ODD AND EVEN DIGITS.