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

   +  FIELD

Which digit does each letter represent?

  • $\begingroup$ 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. $\endgroup$ – r3mainer Feb 1 '16 at 16:44

The solution I got is

(A,F,W,L,E,M,R) = (0,1,9,2,3,6,4) and eight possibilities for (H,I,T,D) being (5,a,7,8), (5,a,8,7), (a,5,7,8), (a,5,8,7), (7,8,a,5), (7,8,5,a),(8,7,a,5), (8,7,5,a).

The methodology is

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.

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