Let the numbers "to carry" be t, u, v, and w that I've written above
the addition:
tuvw
SEND
+MQRE
-----
MQNEY
The numbers "to carry" are either 0 or 1. Since the letter O looks like the
digit 0 (zero), I will use Q instead to avoid confusion.
Obviously t = M = 1, so we have
1uvw
SEND
+1QRE
-----
1QNEY
Case 1: u=1
11vw
SEND
+1QRE
-----
1QNEY
Then S = 8, Q = 9
11vw
8END
+10RE
-----
10NEY
This makes v=1 to prevent E=N
But in order to cause a carry of 1,
E could only be 9, causing N=0
but Q is already 0. That rules
out case u=1.
So u=0
10vw
SEND
+1QRE
-----
1QNEY
so S=9, Q=0
10vw
9END
+10RE
-----
10NEY
v=1 to prevent E=N
101w
9END
+10RE
-----
10NEY
E + 1 = N
w + N + R = 10 + E
D + E = Y + 10w
Try w=0
E + 1 = N
N + R = 10 + E
D + E = Y
Substitute E + 1 for N in the second equation
E + 1 + R = 10 + E
That simplifies to
R = 9
But S = 9,
so w = 1
1011
9END
+10RE
-----
10NEY
N + R = E + 9
1 + E = N
D + E = Y + 10
Substitute 1 + E for N in the 1st equation
1 + E + R = E + 9
R = 8
1011
9END
+108E
-----
10NEY
E + 1 = N
D + E = Y + 10
0 and 1 are used, so Y > 1, so
D + E > 11
So the only possibility is
{D,E} = {6,7} or {D,E} = {5,7}
Try {D,E} = {6,7}
So Y = 3
1011
9END
+108E
-----
10NE3
But E has to be 1 more than E
E can't be 7 because that would make N=8=R.
E can't be 6 because that would make N and D
both be 7.
So
{D,E} = {5,7} which makes Y = 2
1011
9END
+108E
-----
10NE2
Since N is 1 more than E, E can't be 7,
for that would make N=8=R, so E=5, and D=7
1011
95N7
+1085
-----
10N52
So N=6
1011
9567
+1085
-----
10652
Whew!
Edwin
