12.129 Given that all the diagonals of a convex polygon are of equal length, find all possible values for the number of sides of the polygon.

 untranslated text remains unchanged:
已知一个凸多边形的所有对角线的长度都相等,求多边形边数的所有可能值. 

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12.129 Given that all the diagonals of a convex polygon are of equal length, find all possible values for the number of sides of the polygon.
[Solution] Clearly, the diagonals of a square and a regular pentagon are all equal in length.
On the other hand, when $n \geqslant 6$, let the vertices of the convex $n$-gon be $A_{1}, A_{2}, \cdots, A_{n}$ in sequence. Consider the convex quadrilateral $A_{1} A_{3} A_{4} A_{6}$, then we have $A_{1} A_{4}+A_{3} A_{6}>A_{1} A_{3}+A_{4} A_{6}$. Since these 4 segments are all diagonals of the convex polygon, there must be two diagonals of unequal length among them.
Therefore, the number of sides of the polygon that meets the requirement in the problem is 4 or 5.