256 Answersagain giving the digital roots shown. Now, as shown in Amusements in
Mathematics, the digital root of every square number must be either 1, 4, 7,
or 9, so that the required numbers must have the roots 4, 7, 7, to be a square.
The two 7's may be selected in three different ways. But if the fifth number is
included, the total of the three will end in 189 or 389, which is impossible for
a square, as the 89 must be preceded by an even figure or O. Therefore the
required numbers must be: 2,494,651 + 1,385,287 + 9,406,087 = 13,286,025,
which is the square of 3,645.
As illustrating the value of this new method we may be allowed to quote
from the late Professor W. W. Rouse Ball:
"This application is original on Mr. Dudeney's part. Digital properties are
but little known to mathematicians, and we hope his example may serve
to direct attention to the method .... In a certain class of arithmetical prob-
lems it is of great assistance."
- JUGGLING WITH DIGITS
7 + 1 = 8; 9 - 6 = 3; 4 X 5 = 20.
- EQUAL FRACTIONS
The five answers are as follows:2 3 79 3 7 29. 3 9 27.2 3 _ 58 .. 2 _ .6 _ 97
"4 = (; = 158; 6" = 14 = 58' 6" = 18 = 54' 6" = 9" - 174' 1-3" -485'- DIGITS AND PRIMES
The 4, 6, and 8 must come in the tens place, as no prime number can end
with one of these, and 2 and 5 can only appear in the units place if alone.
When those facts are noted the rest is easy, as here shown:47
61
89
2
3
5
207