Consider the constraints of arithmetic when approaching a problem. For example, in the SEND + MORE = MONEY puzzle, two four-digit numbers are added to get a five-digit number. This means that M is a carry. M can only be 1. The constraint that the carry from the addition of two digits can only be 1 starts us on the path to solving the problem.
Update the puzzle as you find solutions for each digit. Having the digits mixed in with the letters makes it easier to solve for new digits. After we find that M = 1 in the SEND + MORE = MONEY puzzle, we would rewrite it as: SEND + 1ORE = 1ONEY
Add the few constraints that are standard with cryptarithmatic puzzles to the constraints inherent in mathematics. One of these constraints is that each letter represents a single digit. We can use this constraint to get the second part of the puzzle to the SEND + 1ORE = 1ONEY puzzle. The letter O is either 0 or 1 because S + 1 or S + carry + 1 = 10 or 11, but S cannot be 1 because is 1; therefore O is 0, and we now have SEND + 10RE = 10NEY.
Determine whether or not there is a carry from one column to find the results of the column to the left. For example, S can be 8 or 9, depending on if there is a carry from the E + 0 = N or E + carry + 0 = N column. E must be different from N, using another constraint. Therefore, we have E + 1 = N, and E and N cannot be 8 and 9 as S must be one of these. With this information, S is 9, and we have 9END + 10RE = 10NEY.
From 9END + 10RE = 10NEY, we see that E + 1 = N with no carry because E cannot be 9 and N cannot be 0. From the next column to the right, N + R + possibly 1 more = E + 10. If we subtract one formula from the other, such that [N + R + possibly 1 more = E + 10] - [E + 1 = N], we get R + possibly one more = 9, which means that R = 8 because S was 9. Now we have 9END + 108E = 10NEY. Continuing in this fashion, we ultimately get 9,567 + 1,085 = 10,652.