Friday, March 9, 2018

Decrypting The CryptArithmatic Problem


What was the 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.
 
Maths Puzzle -  Correct numbers for letters
 
To conclude,
 
  9 5 6 7
+ 1 0 8 5
=========
1 0 6 5 2 

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