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Solution of Erdős Problem 633

Michael Beeson, Miklos Laczkovich, Yan X. Zhang

Abstract

We classify triangles that can be tiled only into a square number of congruent triangles, settling Erdős Problem 633.

Solution of Erdős Problem 633

Abstract

We classify triangles that can be tiled only into a square number of congruent triangles, settling Erdős Problem 633.

Paper Structure

This paper contains 6 sections, 27 theorems, 36 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Forward Direction
  4. Some number theory
  5. Existence
  6. Examples

Key Result

Theorem 1

A triangle $T$ admits a non-square tiling if and only if it satisfies one of the following conditions, where $(A,B,C)$ are the angles of $T$ in some order: $\blacktriangleleft$$\blacktriangleleft$

Figures (15)

  • Figure 1: Every triangle has quadratic tilings.
  • Figure 2: Any isosceles triangle can be cut in half.
  • Figure 3: Some tilings with commensurable angles.
  • Figure 4: $3k^2$ tilings with commensurable angles.
  • Figure 5: A 1215-tiling of an equilateral triangle by $(3,5,7)$.
  • ...and 10 more figures

Theorems & Definitions (58)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3: laczkovich1995tilingslaczkovich2012tilings
  • proof
  • Theorem 4: snover1991
  • Corollary 5
  • proof
  • Theorem 6
  • Proposition 7
  • ...and 48 more