# Wheat field alphametic

Every letter stands for a digit in base-11 representation, different letters stand for different digits:

      WHEAT
+  FIELD
------------
FARMER


Which digit does each letter represent?

• I won't post my brute-force solution here since you tagged this as no-computers, but it found 8 valid solutions to this puzzle. However, the value of FARMER is the same in each case. – r3mainer Feb 1 '16 at 16:44

Let 'a' come after 9 in base-11 and all subsequent numbers I quote are in base 11. We must have W+F+(carried over term from 100s) $\ge$ 10 and here you won't carry over more than a 1 so F=1 and as a result W = 9 or a. It follows that A=0. We must have A+L+(carried over from 1s) = E (mod 11) but cannot have L=E (mod 11). Hence T+D $\ge$ 10 and E=L+1. Since E cannot be 0 or 1, we do not carry anything over to the 100s and 2E=M (mod 11). Hence the possibilities for (L,E,M) are (2,3,6), (3,4,8), (4,5,a), (6,7,3), (7,8,5), (8,9,7). We keep in mind that W=9 or a and the remaining terms must be chosen to satisfy H+I+(carried over)=T+D=R (mod 11). Checking through case by case is reasonably quick on pen and paper and we get the solutions only in the first case.