Problem

Source: Bulgarian Spring Tournament 2023 10.3

Tags: combinatorics



Given is a convex octagon $A_1A_2 \ldots A_8$. Given a triangulation $T$, one can take two triangles $\triangle A_iA_jA_k$ and $\triangle A_iA_kA_l$ and replace them with $\triangle A_iA_jA_l$ and $\triangle A_jA_lA_k$. Find the minimal number of operations $k$ we have to do so that for any pair of triangulations $T_1, T_2$, we can reach $T_2$ from $T_1$ using at most $k$ operations.