## Permutation Math Olympiad Question (Challenging)

March’s Problem of the Month was a tough one on permutations. Only six people solved it! (Site: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm)

The question goes as follows:

In each step one can choose two indices $1\leq k,l\leq 100$ and transform the 100 tuple $(a_1, \cdots, a_k, \cdots, a_l, \cdots, a_{100})$ into the 100 tuple $(a_1, \cdots, \frac{a_k}{2}, \cdots, a_l+\frac{a_k}{2}, \cdots, a_{100})$ if $a_k$ is an even number. We say that a permutation $(a_1, \cdots, a_{100})$ of $(1, 2, \cdots, 100)$ is good if starting from $(1,2,\cdots, 100)$ one can obtain it after finite number of steps. Find the total number of distinct good permutations of $(1, 2, \cdots, 100)$.

The official solution is beautiful and uses induction.

Personally, I used a more brute force technique to get the same answer using equivalence class theory which I learnt in my first year of undergraduate math! It is not so bad in this question, since n is only 100, but for higher values of n the approach in the official solution would be better.

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### 3 Responses to Permutation Math Olympiad Question (Challenging)

1. abyssbrain says:

Haha, I would have also use the equivalence class theory to solve this question if I encountered it. But yeah, it would be ridiculously tedious to solve it if n is higher…

I’ll have a closer look to their official solution, it seems interesting.

Liked by 1 person

• Yeah, their official solution is awesome! It didn’t cross my mind to use the cyclic permutation, so I solved it using a more tedious way.
Thanks for commenting!

Liked by 1 person

• abyssbrain says:

Yeah, that approach was very nice indeed

Liked by 1 person

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