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.

[Illustration:

( 15 Gals )

(31 Gals) (19 Gals)

(20 Gals) (16 Gals) (18 Gals)

]


77.--DIGITS AND SQUARES.

[Illustration: