Оклеивание прямоугольника квадратами с обеих сторон

Дальневосточный математический журнал. 2023. Т. 23. № 1. С. 16–26.

Ожегов Ф. Ю., Дмитриев М. Д.

The paper provides an elementary proof of Kenyon’s theorem that periodic tiling of a plane by squares with periods (1, 0) and (0, λ) is possible only if λ = p±\sqrt{q^2 − r^2} for some rational p≥q≥r≥0. A similar new result is proved about covering of a rectangle with squares from both sides in one layer. The paper also proves a necessary and sufficient condition for covering with equal squares.

Keywords: tiling, замощение