Let $D$ be a disc of diameter 20, and suppose you are given 19 rectangles, each of which is $1 \times 20$. Can $D$ be covered completely by the rectangles? Note that the rectangles can be arranged "any which-way" for the covering.
Note: this was a popular poser in my grad student days; some of us came up with an answer, but had a hard time nailing down the proof. I'd like to hear from anyone with a good approach, or even a good reference, to this question.
Added note: the rectangles are not to be cut up. And the question may be posed differently in terms of 19 width 1 "strips", each infinitely long (i.e. copies of $[0,1] \times R$). In this form the question becomes: Can we cover the diameter-20 disk using 19 of these width 1 strips? [If one could cover via strips, then each strip could be trimmed to a 1X20 rectangle without uncovering any covered points in the disk.]