India, June 23 -- I learnt something new and interesting recently. Most puzzle solvers, including you readers of Problematics, probably know this already, but here it is anyway. To check if a number is a perfect square, one way to proceed is to keep adding its digits until you reach a single digit. If it's a square, the ultimate "digital root" in a single digit must be any one of 1, 4, 7 and 9 - and nothing else.

For example, the digital root of 16 is 1 + 6 = 7, that of 81 is 8 + 1 = 9, that of 49 is 4 + 9 = 13 followed by 1 + 3 = 4, and that of 676 is 6 + 7 + 6 = 19 followed by 1 + 9 = 10 and finally 1. There are, of course, numbers that can have digital roots of 1, 4, 7 or 9 without being perfect squares, for example 70 whose root is 7...