      home puzzles genres studio help about links forum        # Alphametic

Replace the letters in this equation by digits such that it becomes a sound mathematical expression. The same letters are the same digits and different letters are different digits. No number with more than one digit has leading zeros.

## Example

To solve the example puzzle let's first think about what the A can be, occuring at the start of the largest number. An upper bound for the sum of the two smaller numbers would be 99+99=198. Of course that upper bound is too high since not all of the digits A, B and C can be 9, and with a little more thought we could put it at 97+89=186, but 198 is good enough to conclude that A is at most 1, and since it can not be a (leading) zero, it must be exactly 1. Also the other A's can be replaced by a 1.

Next we will concentrate on the value of D. With A being 1 and C at most 9, A+C can be at most 10, but we also have a potential carry of 1 from A+B in case it is larger than 9. AD however can not represent 11, and it must be at least 10, so it is exactly 10 and D is zero.

Now C can be 9 in case A+B does not give a carry, or 8 if it does. Suppose that it does and C is in fact 8, then A+B should be 18 and this is impossible. We can conclude C is 9 and consequently B must be 8.

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