Maximum tilings with the minimal tile property

2020/05/02 by Praton, Iwan
#51F #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2005.00893

Abstract

A tiling of the unit square is an MTP tiling if the smallest tile can tile all the other tiles. We look at the function f(n)=max ∑ s_i, where s_i is the side length of the ith tile and the sum is taken over all MTP tilings with n tiles. If n=k²+3, it was conjectured that f(k²+3)=k+1/k. We show that any tiling that violates the conjecture must consist of at least three tile sizes and has exactly one minimal tile.

Related

• Tilings with the Minimal Tile Property
• Minimal tilings of a unit square
• Tiling a unit square with 8 squares
• Every square can be tiled with T-tetrominos and no more than 5 monominos
• Tiling with Small Tiles