Here's the puzzle. Find substitutions of integers (0-9) for letters for the following subtraction problem. There are no repeated digits (e.g. all the values are not all zero).

                         ANTE
                       - ETNA
                         ----
                         NEAT

To solve this, I first turned it into an addition problem:

                         ETNA
                       + NEAT
                         ----
                         ANTE

First, note that there can be up to 3 carries. A carry can be 0 or 1. Lets call the carries C1, C2, and C3. Each of these can take the value of 0 or 1, but there's only one combination that is correct out of the 8 possibilities: (0,0,0) (1,0,0) (0,1,0) (1,1,0) (0,0,1) (1,0,1) (0,1,1) (1,1,1).

Now we can write four simultanous equations, summing each column, with the carries:

A + T = E + 10*C1
N + A + C1 = T + 10*C2
T + E + C2 = N + 10*C3
E + N + C3 = A

Since it's four equations with 7 variables it's underdetermined, but we know there are only 8 values for C1,C2,C3, so we'll ignore them for now and just deal with N,E,A,T. That leaves us with four equations with four variables. Simple high-school algebra will get the following equation (among others):

5*T - 31*C1 + 11*C2 - 8*C3 = 0

Now we need to choose values for C1,C2,C3 that make T come out to be an integer between 0 and 9. Of the 8 possible combinations to try, it turns out that (1,1,0) is the correct one. It gives T=4.

Back-substituting, gives us the full solution: T=4, A=7, E=1, N=6

                         7641
                       - 1467
                         ----
                         6174

Q.E.D.