Digits in disguise

A cryptarithm is a sum with the digits replaced by letters. The rules:

• each letter stands for a digit;
• no two letters stand for the same digit;
• the first digit of a number is never 0.

They look like guessing games, but they aren't. Every step is reasoning about size, parity and carrying.

Take PQ × 8 = RS: a two-digit number times 8, giving another two-digit number. Guna thinks aloud: '10 × 8 = 80. But the units digits of 10 and 80 are the same, which we don't want', because Q and S would both be 0, and different letters must be different digits.

'For the same reason PQ cannot be 11 as P and Q correspond to different digits.' Next: 12 × 8 = 96. P = 1, Q = 2, R = 9, S = 6: all four different. It fits.

Could PQ be 13? 13 × 8 = 104, which has three digits. And every two-digit number above 12, times 8, gives a three-digit number. So 12 is the only answer, and we've proved it rather than just found it.

Now BYE × 6 = RAY. Anshu starts at the top: 'Since the product is a 3-digit number, B can't be 2 or more. If B = 2, i.e., 2 hundreds, the product will be more than 1200.' And B can't be 0, since it starts a number. So B = 1.

Then Y: 'Y cannot be 7 or more because, if Y = 7, then 170 × 6 = 1020; but we want a 3-digit product.' And 'Y will be even', because Y is the units digit of RAY, which is 6 times something, and anything times 6 ends in an even digit.

That's the whole craft: squeeze each letter from above and below until only one digit survives. Patterns, properties and reasoning, never trial and error alone.