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Solve this equation:  SEND+MORE=MONEY

Each letter represents a different digit. Solve this equation: 

       S E N D
     + M O R E
    -----------
     M O N E Y

This is a delightful puzzle. There is only one answer, but there are several ways to get to the answer. In this article, I will give you one such way to figure out (deduce) what each digit must be.

I urge you to get out your paper and pencil, and go through the process each step of the way with me. Don't just read the message to find the answer, do the work, and, if you can, stop reading this article and try to figure out what I'll do next before I do it.

There are other cryptarithms like "USSR+USA=PEACE" that you might enjoy trying to solve. I'm sure that your teacher knows many more. I will try to post some of them to http://www.mazes.com/Q-and-A.html.

Okay, get out your paper and pencil and let's get started: Feel free to stop at any time and try to figure out the rest of the problem by yourself. You won't hurt my feelings.

Remember, the rules for this kind of puzzle: 

       S E N D
     + M O R E
    -----------
     M O N E Y

First, figure out what the letter M stands for. Hint: What is the largest possible answer for S+M (in the thousands column)?

Even if you have a carry from the hundreds column, the largest possible numbers for S and M are 9 and 8. 9+8+1 = 18. This tells us that M must be equal to 1. Now, we have: 

       S E N D      Digits you haven't used yet
     + 1 O R E        0, 2, 3, 4, 5, 6, 7, 8, 9
    -----------
     1 O N E Y

What can we tell about the letter S and the letter O (thousands column)?

There might be a carry from the hundreds column. If so, it will be a carry of 1. There might be no carry (zero). Let's look at both of these possibilities Algebraically:

with "carry" of 1

with no carry

S + 1 + carry(1) = 10 + "O"

S + 1 = 10 + "O"
simplify (1+1=2)

S + 2 = 10 + "O"
subtract 2 or 1
from both sides

S + 2 - 2 = 10 - 2 + "O"
S = 8 + "O"

S + 1 - 1 = 10 - 1 + "O"
S = 9 + "O"
Conclusions:

If "O" is 0, S = 8
If "O" is 1, S = 9

If "O" is 0, S = 9

As you can see, S must either be 8 or 9.

Let's assume that S is equal to 8 and see what happens. If this doesn't work, we'll try S=9:

As we concluded above, if S is 8, then O is zero, because 8+1+carry must equal 10. Let's see what it looks like: 

       8 E N D
     + 1 0 R E
    -----------
     1 0 N E Y

And because this answer requires a carry, E + 0 + carry? must equal 10 + N. Let's look at this Algebraically, both ways, with or without a carry.

with "carry" of 1

with no carry

E + 0 + carry(1) = 10 + N

E + 0 = 10 + N
simplify (0+1=1)

E + 1 = 10 + N

E = 10 + N
subtract 2 or 1
from both sides

E + 1 - 1 = 10 - 1 + N
E = 9 + N

E = 10 + N
Conclusions:

If N=0, E=9
but N cannot be zero

If N=0, E=10
but "O" = 0

Impossible: N cannot be zero because "O" is already zero

We have just proved that S cannot equal 8, so S must equal 9. Here's what we have: 

       9 E N D
     + 1 O R E
    -----------
     1 O N E Y

What are our possibilities for 9+1+?

From the above, we know that O must be either zero or one, BUT M is already equal to 1, so O <> 1. This means that "O" must be equal to zero. Here's what we have now: 

         1
       9 E N D
     + 1 0 R E
    -----------
     1 0 N E Y

Look at the tens column. N+R must be greater than nine, because we MUST have a carry over to the hundreds column.

Why must we have a carry? Because E+0=N. E must not be equal to N, so there must be a carry, which we know has to be 1. We can say this Algebraically as:

E + 0 + carry = N
or
N = E + 1

And, as I said, above, N+R must have a carry, so:

N + R + carry? = 10 + E

Is there a carry or not in N+R=10+E?

with "carry" of 1

with no carry

N + R + 1 = 10 + E

N + R = 10 + E
subtract 1

N + R = 9 + E

Substitute:
N = E + 1

E+1 + R = 9 + E

E+1 + R = 10+E
subtract E
from both sides

1 + R = 9

1 + R = 10

subtract 1
from both sides

R = 8

R = 9

Possible?

Only Conclusion

Impossible because
M already is 9

R must equal 8

Since R must equal 8, let's see what we have now? 

       0 1 1
       9 E N D
     + 1 0 8 E
    -----------
     1 0 N E Y

Again, let's use a little bit of Algebra and solve for two simultaneous equations, one equation for the tens column and one for the hundreds column. Since we have two equations and two unknowns, we should be able to get both values, E and N.

hundreds column

tens column

1+E+0=N

1 + N + 8 = 10 + E
simplify

1 + E = N

9 + N = 10 + E
substitute
N= E+1

9 + E+1 = 10 + E
simplify

10 + E = 10 + E
subtract E
from both sides:

10 = 10
that doesn't help.

Okay, that was a waste of time. That's one case where solving two simultaneous equations doesn't help us. So we don't have to look up above again, here's what we know so far: 

       0 1 1
       9 E N D      Digits still available?
     + 1 0 8 E      2, 3, 4, 5, 6, 7
    -----------
     1 0 N E Y

What are the allowable values for E?

We know that E can only be 2, 3, 4, 5 or 6. It cannot be 7 because E+1=N, and N cannot equal 8 (R=8).

We also know that D+E is greater than 11 because Y cannot be 0 or 1 (O and M took care of those values).

Why don't you experiment with values of E and figure out what N and D and Y must be equal to. We know that:

What do we have now? 

       0 1 1
       9 E N 7      Digits still available?
     + 1 0 8 E      2, 3, 4, 5, 6
    -----------
     1 0 N E Y

Okay, I'm going to let you finish the work from here. What do you know?

What did you come up with?

How did we solve this problem? Sometimes we used trial and error; sometimes we used our deductive and logical abilities; and sometimes we used Algebra. We could have done the whole thing without Algebra, but I'll let you figure out how to do it that way (if you're interested). Actually, though, even if you don't use Algebra, you sort of still are using it, even if you don't know it.

If all you have done is come down this far to get the answer, I urge you to go back to the beginning and actually try every step with me, with your paper and pencil. You will learn more from actually doing the work.

And, of course, you MUST

Good luck.

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