Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

Abstract

AbstractA well-known open problem of Meir and Moser asks if the squares of sidelength 1/n for $$n\ge 2$$ n ≥ 2 can be packed perfectly into a rectangle of area $$\sum _{n=2}^\infty n^{-2}=\pi ^2/6-1$$ ∑ n = 2 ∞ n - 2 = π 2 / 6 - 1 . In this paper we show that for any $$1/2<t<1$$ 1 / 2 < t < 1 , and any $$n_0$$ n 0 that is sufficiently large depending on t, the squares of sidelength $$n^{-t}$$ n - t for $$n\ge n_0$$ n ≥ n 0 can be packed perfectly into a square of area $$\sum _{n=n_0}^\infty n^{-2t}$$ ∑ n = n 0 ∞ n - 2 t . This was previously known (if one packs a rectangle instead of a square) for $$1/2<t\le 2/3$$ 1 / 2 < t ≤ 2 / 3 (in which case one can take $$n_0=1$$ n 0 = 1 ).