If each letter is a unique representation of a digit, and each word is a square integer, what are these four numbers?

Source: American Math Monthly 1956

By comparing squares with exactly 2 repeated digits in position 3 and 4 from 10000 to 100000 with squares with no repeated digits from 1000 to 10000 (excluding those with zeroes in the tens and hundreds positions) it is possible to generate a reasonably small list where the first digit, M,in the first set is matched with the second digit, M in the second set. Starting with the candidates where there is only one entry in one of the sets, 34225 was in fact the second candidate I tried, giving:

A E L M O R S T X Y
9 4 0 3  1 2 6 8 7 5

where MERRY = 34225 =185^2, XMAS = 7396 = 86^2, TO = 81 =9^2, and ALL = 900 = 30^2.

Since this is a valid solution, I assumed uniqueness and did not look for others.

Edited on July 24, 2018, 1:05 am

Posted by broll on 2018-07-24 01:02:24