An "almost square" is a rectangle with sides which are consecutive integers. Almost squares 1 × 2, 2 × 3, 3 × 4, . . . n × (n+1) can be tiled inside another almost square when n = 1, 3, 8, and 20, as the pictures below show.

In January of 2012, I heard from Daan van den Berg that he and his students Florian
Braam, Mark Moes, and Emiel Suilen, had found a solution for n=34, the only other possible case. They also sent me this link to all the solutions for n=20.

n=1

n=3

n=8

n=20

In February of 2013, Giovanni Resta sent me the tiling below, which he says he found in January of 2007. The widths are given, and the + or – sign indicates whether the height is 1 larger or smaller.

n=34