Problem:
Let A1​,A2​,A3​,…,A12​ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set {A1​,A2​,A3​,…,A12​}?
Solution:
Each pair {Ai​,Aj​} of the (212​)=66 pairs of vertices generates three squares, one having diagonal Ai​Aj​, and the other two having Ai​Aj​ as a side. However, each of the three squares A1​A4​A7​A10​,A2​A5​A8​A11​, and A3​A6​A9​A12​ is counted six times. The total number of squares is therefore 3⋅66−15=183​.
The problems on this page are the property of the MAA's American Mathematics Competitions