Hypercubic Self-Tilings

TL;DR

This paper addresses hypercube tiling problems, specifically improving bounds on the minimal number of hypercubic tiles needed to tile higher-dimensional hypercubes, with a focus on the tesseract.

Contribution

It improves the known bound for tiling a tesseract with hypercubic tiles from 808 to 733, advancing understanding of hypercube tiling constraints.

Findings

Bound for tiling a tesseract reduced from 808 to 733

Provides insights into hypercube tiling in higher dimensions

Addresses longstanding problem in number theory and geometry

Abstract

In 1946 Fine and Niven posed problem E724, asking to demonstrate that every hypercube can be tiled by any number of hypercubic tiles larger than some value. This requires only basic number theory, but the problem of finding the smallest such number is much more involved. For the square this is known to be 5, and the cube 47. No other values are known. This paper improves the bound for tesseracts from 808 to 733.