Mathematical_Recreations-Kraitchik-2e

80 Mathematical Recreations
from 1931 to 1939, many problems concerning the restoration
of mathematical operations. The name is due to Minos
(Vatriquant) who first used it in Sphinx in May, 1931.
Minos prefaced his problem with the following remark:
"Cryptographers ... put figures in place of letters. By way
of reprisal we have put letters in place of figures. " A" cryptarithm"
is, then, an arithmetical operation - such as an
addition, a multiplication, or a division - in which the digits
have been replaced by letters or other symbols, and one is
challenged to find the original numbers.
Although Minos originated the name, problems of this
sort were known earlier, and there are better examples in
"La Mathematique des Jeux" (1930). Here are two.
1. I was sitting before my chess board pondering a combination
of moves. At my side were my son, a boy of eight,
and my daughter, four years old. The boy was busy with his
home work, which consisted of some exercises in long division,
but he was rather handicapped by his mischievous sister,
who kept covering up his figures with chess men. As I
looked up, only two digits remained visible. I have put down
the division in Figure 10 just as I saw it. Can you calculate
the missing digits for my little boy without annoying his
sister by removing the chess men?
Solution: This is particularly easy. First, observe that
the five-digit quotient forms only three products with the
divisor. Therefore two of the five digits must be O's. These
cannot be the first or last, since both obviously form products.
They are therefore the second and fourth digits, those
covered by the white bishops. Furthermore, the two-digit
divisor, when multiplied by 8, gives a two-digit product; but
when multiplied by another number, the one concealed under
the first white rook, it gives a three-digit product; therefore
the hidden multiplier must be larger than 8, consequently 9.
Both the first and last digits in the quotient give three-digit