aime26-06

Proposed solution

To solve this problem, we need to find the number of functions \(\pi\) mapping the set \(A = \{1, 2, 3, 4, 5, 6\}\) onto itself such that \(\pi(\pi(\pi(\pi(\pi(\pi(a))))) = a\) for every \(a \in A\).

First, let's analyze the given condition: \(\pi(\pi(\pi(\pi(\pi(\pi(a)))) = a\). This implies that applying \(\pi\) five times returns the original element \(a\). Therefore, \(\pi\) must have an order of 5. In other words, \(\pi\) must be a permutation of \(A\) that consists of cycles whose lengths divide 5.

The possible cycle lengths that divide 5 are 1 and 5. Since \(\pi\) maps \(A\) onto itself and is onto, every element must be included in a cycle. Thus, the only possible cycle structure for \(\pi\) is a single cycle of length 5.

Next, we need to determine how many such 5-cycles exist. A 5-cycle is a permutation where all elements are in a single cycle of length 5. The number of ways to arrange 5 elements in a cycle is given by \((5-1)!\), because we can fix one element and permute the remaining 4 elements in any order.

Calculating this, we have:
\[
(5-1)! = 4! = 24
\]

Thus, there are 24 such functions \(\pi\) that satisfy the given condition.

Therefore, the number of functions \(\pi\) mapping the set \(A\) onto itself such that \(\pi(\pi(\pi(\pi(\pi(\pi(a)))) = a\) for every \(a \in A\) is \(\boxed{24}\).