We cannot remove two touching coins; if you did, any third coin
originally making a triangle with then would then be slideable. We
are able to remove the centre and corner coins without enabling any
of the remaining coins to slide.If we do this, all of the coins
touched at least one of the removed coins so we cannot remove any
If we remove a coin which is not one of these four coins first, it
would be one of a middle pair of an edge. Four coins touch this
coin, leaving five possible coins to remvoe. Four of the five coins
would lie along an edge of the frame, with two of these forming a
triangle with the fifth. Clearly, only one of the three coins which
form a triangle may now be removed, together with only one of the
other two coins as they are touching. So we conclude that the
maximum number of coins which may be removed is four.
This problem is taken from the UKMT Mathematical Challenges.