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Matching Numbers and Letters

A group of five logical thinkers each push a different button on a machine
(numbered 1, 2, ..., N). When the fifth button is pushed, five balls are
released (lettered A, B, C, ...), each corresponding to one of the pressed
buttons. After four such rounds, they are able to uniquely identify which
number is linked to which letter. What is the highest possible number on
the machine?

Extension: Is there a way to determine the maximum number of
buttons for P people in R rounds? Or the minimum number of rounds needed
to identify N pairings?

Source: Original. Based upon puzzle 12 in the High IQ Society's Smartest Person Challenge 2006.

Solution

Mail to Ken