Pick's theorem
A simple polygon has all of its vertices at points with integer coordinates. It has integer points in its interior and integer points on its boundary. Find a formula for its area in terms of and , and prove it.
History and references
Solved in 1899 by Georg Alexander Pick.
Georg Pick published the formula in 1899, and it became widely known in English after Hugo Steinhaus included it in the 1950 edition of his book Mathematical Snapshots. One proof checks it for triangles and shows that both sides behave the same way when two polygons are glued along an edge. Another triangulates the polygon into lattice triangles with no other lattice points and combines Euler's formula with the fact that each such triangle has area 1/2. There is no direct analogue in three dimensions, since the Reeve tetrahedra all have the same four lattice points as their only lattice points but different volumes.
References
Solutions 0
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