**possible
related topics** area, circles

Learners could start with all the pens the same size (say, unit diameter) and vary the diameter of the cup, checking each time how many will fit in. Starting with a cup diameter of 1 and increasing by 1 each time, they will find the sequence 1, 2, 7, 11, 19, ... (sequence A023393 at The On-Line Encyclopedia of Integer Sequences).

Smaller pens can of course be fitted into the gaps leftover, but then it becomes much more complicated!

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© Colin Foster 2012, www.foster77.co.uk