### Decrypting The CryptArithmatic Problem

Let's take a look at the equation once again.
```       S  E N D
+   M O R E
-----------------
= M O N E Y```
` `
`Now letter M must be representing the carry generated & it must be 1. `
` `
`And if M = 1 then S must be 9 or 8 if carry is generated from hundreds place. `
`In any case, O can be either 0 or 1. But can't be 1 as M = 1 hence O = 0.`
` `
`If O = 0 then E + 0 = N i.e. E = N if there is no carry from tens place. `
`Hence, N = E + 1.`
` `
`Let's turn towards tens place now. With no carry from units place N + R = 10 + E .`
`Putting N =  E + 1, we get, R = 9. And with carry from units place, we have,`
`1 + N + R = 10 + E, gives us R = 8. Hence either R = 9 and S = 8 or R = 8 or S = 9.`
` `
`For a moment, let's assume R = 9 and S = 8 with no carry from units place, then, `
` `
```   8 E N D
+ 1 0 9  E
=========
1 0 N E Y ```
` `
`For this to be correct, we need carry at thousands place generated from hundreds `
`place. That's only possible if E = 9 and carry is generated from tens place`
`forwarded to hundreds place. Since 9 is already being used for R this combination`
`is just impossible.`
` `
`Hence, R = 8 and S = 9 with carry 1 from units place.`
` `
`Now it looks like,`
` `
```   9 E N D
+ 1 0 8 E
========
1 0 N E Y ```
` `
`This means D + E >= 10 i.e. D + E = 10 + Y.  And numbers left are 2,3,4,5,6,7.`
` `
`If E = 2 then D must be 8 or 9 for D + E >= 10.`
`Since 8 and 9 already taken, this is not possible.`
` `
`If E = 3 then D can be 7 but Y would be 0 in the case. `
`Since O = 0 already taken this value of E is also not valid. `
`For any other value of D, D + E < 10 .`
` `
`If E = 4 then D = 7 or 6 and Y = 0 or 1.`
`Both are taken hence this value of E in invalid.`
`Also,D <= 5 in the case gives  D + E < 10 .`
` `
`If E = 6, N = 7 then, D <= 5. `
`With D = 5 or 4, Y = 0 or 1, both are used for O and M already.`
`And for D = 2 or 3,D + E < 10 .`
`In short, this value of E is also not valid.   `
` `
`So only value of E left is 5. Hence, N = 6 and D = 7. That gives, Y = 2.`
` `
` `
`To conclude,`
` `
```  9 5 6 7
+ 1 0 8 5
=========
1 0 6 5 2 ```
```
```