\section*{Problem \(4-340934=341034\)}

A square \(A B C D\) is divided into 25 congruent smaller squares.

Let \(n\) be a positive integer with \(n \leq 25\), and let \(n\) different colors be chosen, with 25 tiles of the size of the smaller squares provided for each of these colors.

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From these \(n \cdot 25\) tiles, 25 are to be selected and placed on the square \(A B C D\) such that each smaller square is covered by exactly one of the selected tiles.

A number \(n\) will be called a "friendly" number if and only if the following holds:

For any selection of 25 of the \(n \cdot 25\) tiles, where each of the \(n\) colors is represented by at least one tile, it is possible to arrange the distribution on the smaller squares such that the covered square \(A B C D\) forms a symmetric pattern with respect to the line through \(A\) and \(C\). Determine all "friendly" numbers among the positive integers \(n \leq 25\)!
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For every symmetric arrangement, it holds that the 10 sub-squares above the diagonal \(A C\) must each have the same color as the 10 sub-squares below this diagonal, while the colors of the 5 diagonal squares can be arbitrary. This is also sufficient. Therefore, a selection of 25 tiles cannot produce a symmetric arrangement with respect to \(A C\) if it does not contain 10 pairwise disjoint pairs of tiles of the same color.

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be the number of tiles of colors \(1, 2, \ldots, n\) among the 25 selected. Then, the maximum number of such pairs that can be formed is

\[
P:=\left[\frac{a_{1}}{2}\right]+\left[\frac{a_{2}}{2}\right]+\cdots+\left[\frac{a_{n}}{2}\right]
\]

Given \(a_{1}+a_{2}+\cdots+a_{n}=25\), we have \(P=\frac{25-u}{2}\), where \(u\) is the number of odd numbers among \(a_{1}, a_{2}, \ldots, a_{n}\). (Since 25 is odd, \(u\) is also odd, making \(P\) an integer.) Therefore, a symmetric arrangement with respect to \(A C\) cannot be formed from these tiles if there are more than 5, i.e., at least 7, odd tile counts among \(a_{1}, a_{2}, \ldots, a_{n}\).

For this to be the case, \(n \geq 7\) must hold. This means that all \(n \leq 6\) are "friendly." However, if \(n=7\), one can choose \(a_{1}=a_{2}=\cdots=a_{6}=1\) and \(a_{7}=19\), or for \(n>7\), finally \(a_{1}=a_{2}=\cdots=a_{7}=1\) (and the rest arbitrarily \(\geq 1\)), such that no symmetric arrangements with respect to \(A C\) are possible with this selection. Therefore, all \(7 \leq n \leq 25\) are "unfriendly."