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This is a cryptarithm puzzle I found in a mathematical contest in 2004. The questions were available on the website of the of the AQJM (in french, Association Québécoise des jeux mathématiques) and we had about two months to give our answer.

$$\text{NINE} \times \text{THREE} = \text{NEUF} \times \text{TROIS} $$ What is the result of the multiplication?

Also, we know that $\text{TROIS}$ and $\text{NINE}$ are multiple of 3 and that $\text{NEUF}$ and $\text{THREE}$ are multiple of 9.

The usual rules of the alphametic puzzle apply here.

  • The same letter always represent the same number
  • Two letters represent different numbers
  • Numbers can't start with a 0

Since the questions were on the internet, we could use whatever means we want to find the answer. A friend of mine created a small programm to test all possible combinaison. I did it with pencil and paper and it took me about 10 hours of work to find the answer, mostly by trials and errors to exclude possiblities. I wonder if someone has a better way of tackling such problem.

Complete contest available here

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Without the no-computers tag, I just did a search and found the unique solution:

9096 x 13266 = 9648 x 12507

It's slightly imperfect that NINE is not a multiple of $9$. But one cannot expect more... this already is a quite funny identity.

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  • 3
    $\begingroup$ The OP already knew the answer. He is asking for a smart way to solve it without computers. $\endgroup$ – Jens Nov 28 '19 at 19:36
  • $\begingroup$ Looking at some of the old problems in that contest, I think they don't always give problems with a "smart" solution. Sometimes a bruteforce search is inavoidable. $\endgroup$ – WhatsUp Nov 28 '19 at 22:36
  • $\begingroup$ @WhatsUp Sadly the contest only gives answer, and not solution. Usualy the problems are solvable by pen and paper only. I've seen complete solutions of other exam of the same contest and there usually a simple way to solve the problems. +1 for the answer since it might help others. $\endgroup$ – Alain Remillard Nov 29 '19 at 14:49

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