# How do you solve cryptarithms?

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Let me give you an example. Probably the most famous cryptarithm:

*send+more=money*

In cryptarithms like this it's good to search for digits 0 and 9 first. Since *money* is one digit longer than *send* and *more *it follows that first digit *m* must be 1 that is "carry 1". So now we have:

*send+1ore=1oney*

Here we have *s+1=10* or *s+1=11 *in the later case we would have *o=1* which cannot be true since we have already establish that *m=1*, thus *o=0*. Now we have:

send+10re=10ney

Here we have *s+1=10* so *s* can only be equal to 9 or 8. But if *s=8 *that would mean that we have "carry 1" from addition of previous digits namely *e+0=10* (we can have at most 10 only if *e=9* and we have "carry 1" from previous addition) which cannot be because we would have *n=0* but we have already establish that *o=0*. Thus we have *s=9*.

9end+10re=10ney

Now we have *e+0=n* meaning that *n* is by one greater than *e* i.e. *n=e+1 (1)*

Also because we must have "carry 1" from previous digits we have *n+r+(1)=e+10 (2)*

(1) is in brackets because we still don't know if we have "carry 1" from previous digits). Now we subtract (2)-(1) and get

*r+(1)=9*

*A*nd since we've establish that *s=9,* *r* cannot be 9 so it must be *r=8*. And we now know that we have "carry 1" from previous (first) digits.

*9end+108e=10ney*

Now for the fist digits *d+e>12* (because we have "carry 1" so it must be >10 and y cannot be 0 or 1 because *m=1* and *o=0*). So what digits can we have for *d* and *e?* We can't have 9 or 8 (*s=9* and *r=8*). We can only have 7 and 6 or 7 and 5. Now we remember equation (1) and see that *n* lies between *d* and *e*. Hence we have *e=5, n=6* and *d=7*. And also *d+e=10+y* gives us *y=2*. So the solution is

9567+1085=10652

which is true.

There is no general way for solving such problems but you usually start by searching for 0, 1 or 9 and after that you try some logic and basic arithmetic and if you must trial and error.

How do you solve cryptarithms?

Cryptarithms are a type of mathematical puzzle in which the digits are replaced by symbols (typically letters of the alphabet). For example:

**9567 + 1085 = 10652**

can be represented like this:

**abcd + efgb = efcbh**

Cryptarithm Rules

- Each letter represents a unique digit.
- Numbers must not start with a zero.
- The solution is unique (unless otherwise stated).

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