OK, here's how I
solve this type of
problem. It also shows you that there's just this one solution.
SEND
+ MORE
======
MONEY
We have to find values for S,E,N,D,M,O,R,Y (
8 digits out of
10). Now, we're adding
2 4-digits
numbers. Since
9999+9999 < 20000, M cannot be >=
2. And by the "usual rules" for this kind of question, it can't be 0. So
M=1.
Now, looking at the fourth (left-most) column, we have either S+1>=10 (if there's no carry) or 1+S+1>=10 (if there's carry). So S=8 or 9, and O=0 or 1. Since 1 is already taken, O=0 (just as well that, or the typography would get confusing).
In the third column, we can't have E+0=N (no carry), so E+1=N and there's carry from the second column. So in the second column either N+R=10+E=9+N, and R=9, or there's carry and 1+N+R=10+E=9+N, and R=8. So R=8 or 9, just like S.
IF S=8 and R=9,
we're looking at
8END
+ 109E
======
10NEY
But this cannot possibly work: we need to get either E+0=10+N or 1+E+0=10+N in the third column, to get carry in the fourth column. Neither is possible (we've already used up both 9 and 0). So...
S=9 and R=8.
We're looking at
9END
+ 108E
======
10NEY
We've already used up the digits 0,1,8,9, and
N=E+1, so the only choices for E are 6,5,4,3,2.
We know we must have carry from the first column into the second, so D+E>=10. D is at most 7, so we immediately rule out E=2 (7+2<10). Also E=3 is impossible (because then either D=7 and Y=0=O, or D<7 and E+D<10, both of which are impossible).
If E=4, then D=7 or D=6 don't work (because then Y=1 or Y=0, and both are already taken), and D<6 doesn't work because then E+D<10.
If E=6 then N=7, so D<=5. But D=5 yields Y=1 and D=4 yields Y=0, both taken, and D<=3 gives E+D<10.
So E=5 and N=6. D=7 (the alternative, D<=4, is again too small), so Y=2 and the solution is
9567
+ 1085
======
10652
We also see there are no other solutions to this
cryptarithm.