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I have a funny aphametic problem:

  I A
+ I A
-----
  P B

I have to find which are the values of the letters. I know I can try backtrack with forward research, MRV heuristic and degree heuristic. Yet I'm not sure how to apply them on cryptarithms/alphametrisms.

My attempt

I tried to find the letters values with backtrack research.

Domains

  • $D(A) = D(I) = \{1,..., 9\}$
  • $D(P) = D(B) = \{0,..., 9\}$

Constraints

  • $10(I + I) + (A + A) = 10P + B$

So I did iterated step by step by hand on the values:

  1. $A= \{I = 1\} \rightarrow$ consistant
  2. $A = \{I = 1, A = 0\} \rightarrow$ consistant
  3. $A = \{I = 1, A = 0, P = 0\} \rightarrow$ inconsistant
  4. $A = \{I = 1, A = 0, P = 1\} \rightarrow$ inconsistant
  5. $A = \{I = 1, A = 0, P = 2\} \rightarrow$ consistant
  6. $A = \{I = 1, A = 0, P = 2, B = 0\} \rightarrow$ inconsistant
  7. $A = \{I = 1, A = 0, P = 2, B = 1\} \rightarrow$ inconsistant
  8. $A = \{I = 1, A = 0, P = 2, B = 2\} \rightarrow$ inconsistant
  9. $A = \{I = 1, A = 0, P = 2, B = 3\} \rightarrow$ inconsistant car $A+A = 0 \neq 3$

    ... until we change the value from A to 2 (because I already equal to 1) so far I don't know the answer.

It seems it's going to take ages. Am I doing it right ?

Maybe the following graph might be of any help ?

graph of AI + AI = PB

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