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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.
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.
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?