Associated Topics || Dr. Math Home || Search Dr. Math

Finding the Digits of SEND + MORE = MONEY

```
Date: 05/19/98 at 23:34:59
From: Echo
Subject: A Basic Story Problem Of Guessing Mystery Numbers

Here is one of the more well-known cryptic problems, so I have been
told, but I am stuck. I really don't know quite what to do or where
to start. If this is one of the problems in your archives, please let
me know. I looked, but was unable to find it.

SEND
+ MORE
------
MONEY

I am trying to find what each of the letters represents. Thanks so
```

```
Date: 05/20/98 at 13:04:05
From: Doctor Rob
Subject: Re: A Basic Story Problem Of Guessing Mystery Numbers

These problems can be reduced to systems of equations and
inequalities:

(1)  D, E, M, N, O, R, S, Y different whole numbers
(2)  0 <= D <= 9
(3)  0 <= E <= 9
(4)  1 <= M <= 9    (it's the leading digit of "MORE", so not zero)
(5)  0 <= N <= 9
(6)  0 <= O <= 9
(7)  0 <= R <= 9
(8)  1 <= S <= 9    (it's the leading digit of "SEND", so not zero)
(9)  0 <= Y <= 9
(10)  D + E = Y + 10*a     (units column, carry a)
(11)  N + R + a = E + 10*b (tens column, carry b)
(12)  E + O + b = N + 10*c (hundreds column, carry c)
(13)  S + M + c = O + 10*d (thousands column, carry d)
(14)  d = M                (ten-thousands column)
(15)  0 <= a <= 1
(16)  0 <= b <= 1
(17)  0 <= c <= 1
(18)  0 <= d <= 1
(19)  a, b, c, d are whole numbers

First of all, from (4), (14), and (18), 1 <= M = d <= 1, so:

(18a) d = 1
(4a)  M = 1

Now, in (13), S + 1 + c = O + 10.  If c = 0, by (8), (13), and (6),

9 >= S = O + 9 >= 9

so S = 9 and O = 0. If c = 1, 9 >= S = O + 8 >= 8, so either S = 8 and
O = 0, or S = 9 and O = 1. By (1) and (4a), O cannot be 1, so in
either case:

(6a)  O = 0
(13a) S + c = 9

Looking at (12), (3), and (16):

9 = 8 + 1 >= E + b = N + 10*c > 10*c

so c < 9/10, and by (17) and (19):

(17a) c = 0

and by (13a):

(8a)  S = 9

From (12), (6a) and (17a), E + b = N, so, by (1):

(16a) b = 1
(12a) E + 1 = N

Now, from (11) and (16b), N + R + a = 10 + E. Putting this together
with (12a):

(11a) R + a = 9

which tells you by (1), (15), and (8a) that:

(7a) R = 8
(15a) a = 1

From (10) and (15a):

(10a) D + E = 10 + Y

Now, consider the possibilities for Y. It must be 2, 3, 4, 5, 6, or 7.
Use (10a) and (12a) along with (1), (4a), (6a), (7a), and (8a).

So far we have:

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

From here, you can finish by yourself. For more assistance, refer to
this answer from the Dr. Math archives:

Send More Money
http://mathforum.org/library/drmath/view/57951.html

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Puzzles

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search