Solution
Crossings in a Regular Dodecagon
Show the problem again
All the diagonals of a regular 12-gon are drawn. How many distinct points in the interior of the polygon lie on two or more diagonals? Give an integer.
Worked solution
The answer is 301. In a convex 12-gon in general position, every 4 vertices determine one crossing (of the two diagonals of that quadrilateral), giving $\binom{12}{4} = 495$ crossing pairs. In the regular 12-gon many crossings coincide: a point where $m$ diagonals meet absorbs $\binom{m}{2}$ of these pairs but counts once. Exact computation of all the crossing points shows: 228 points with 2 diagonals, 60 points with 3, 12 points with 4, and the centre, where the 6 long diagonals all meet. Check: $228 + 60 \cdot 3 + 12 \cdot 6 + 15 = 495$ pairs. The number of distinct points is $228 + 60 + 12 + 1 = 301$, i.e. $495 - (60 \cdot 2 + 12 \cdot 5 + 14) = 301$.
Source: Interior intersection points of the diagonals of a regular polygon (Poonen and Rubinstein, 1998); this case computed for AxiomIQ. Statement written for AxiomIQ.