TITLE: Is it possible to cover a $70\times70$ square with $24$ squares with side length $1,2,3\ldots24$?
QUESTION [7 upvotes]: Is it possible to cover a $70\times70$ square with $24$ squares with side length $1,2,3\ldots24$?

REPLY [7 votes]: This problem was posed by Martin Gardner himself in 1966 (Scientific American, Sept. 1996). As far as I know, it remained unsolved until it was manually checked by a computer program in 2002 (link). It is impossible.
Further notes: Consider the general problem: is it possible to tile an $L\times L$ square with $k$ squares of side length $1,2,3\ldots k$? It turns out, the only solution to
$$1^2+2^2+3^2+\cdots+k^2=L^2$$
is $k=24,L=70$ (besides the obvious $k=L=1$). Here is a link to the proof by Watson in 1918 (it's in the first few pages) So this problem is unique, in a way.
Additionally, the general idea of tiling a square completely with other squares is called "squaring the square" and is extensively studied (link).