Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength.

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