need help with cryptic message: send+more=money?

Question

the problem is send+more=money. you have to replace the letters with the numbers 0 through 9 so that the equation will be true. you will only use 8 of those numbers. please tell me the answer and give a thorough explanation on how you got that anwer.

Answer 1

eliminate variables by doing a bit of algebra:

9567 + 1085 = 10652

we know that m=1 since it could only be created by a carry
s+1=1o s=9 so o=0
9end
10re
10ney
e+1=n given a carry from n+r
there has to be a carry from d+e
so n+r+1-10=e
e+1+r+1-10=e
r=8
n=e+1 shows they are sequential

n+r-10=e

9end
108e
10ney
the key is e

e+1=n
n>4 since y>1 and e>1
so n=5,6,or7
with n=6 e=5

956d
1085
1065y
d>5 leaves only 7
y=2
9567
1085
10652

j

Answer 2

the answer is 9567+1085=10652

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.

If S is 8, there MUST be a carry from the prior column, 8 + 1 + carry = 10, which would make "O" equal to zero.
If S in 9, "O" might be zero, if there is no carry; or "O" would be one, if there is a carry.
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+?

9 + 1 + nocarry = 10
M = 1 and O = 0
9 + 1 + carry = 11
M = 1 and O = 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:

N = E + 1
E <> 7
D + E = 12 or 13
Why? remember, we said D+E>11
and the largest possible values for D & E are 6 & 7
What are the only possible #s out of {2,3,4,5,6,7} that add up to 12 or 13?
5+7 = 12
6+6 = 12 (illegal, D<>E)
6+7 = 13
If E<>7, then D must be equal to 7
and E must be equal to 5 or 6.
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?

Y is equal to either 2 or 3
7+E is equal to either 12 or 13
N is equal to E+1
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.

Answer 3

Maxim m you are so brainy, respect!