Radio Times Puzzle for Today No. 272

by Professor Michael Paterson, University of Warwick

Published on Thursday 19 July 2018 (link)

I have a tray of length 5 and width 2 so 10 round coins of width 1 will fit in it snugly without overlaps. No room for another. Similarly, a try of length 50 will accommodate only 100 coins. Things get more interesting with a longer tray! A tray of length 500 and width 2 can accommodate at least 1001 coins. Show how this can be done. To be clear, this is just about fitting non-overlapping circles in rectangles, so no trickery with funny-shaped coins or thermal expansion coefficients! A useful hint is to start packing coins from the middle not from one end.

For a more mathematical challenge, what is the smallest \(n\) so that \(2n+1\) circles of diameter 1 can be packed without overlap in an \(n\) by 2 rectangle? If you think you have a good answer to this, do let me know, but note that the really hard problem, still I think unsolved, is to prove the best optimum result.