Four copies of the P pentomino pictured below can easily be placed in a 5x5 square without overlapping.
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Prove it is impossible to put 5 copies in a 5x5 square without overlapping.
Which of
the other 11 pentominoes can be included with four P pentominoes such that all five can be placed in a 5x5 square?
Each P must be in a corner. If you place the nose in a corner it leaves a square that must be filled by the nose of the other. This leaves a 2x1 gap that cannot be filled. So every corner must be filled by the 2x2 end of the P. Two neighboring Ps cannot face towards each other (no room) and if they face away from each other they leave a 2x1 gap. So the Ps in the 4 corners must make a ring. But the hole left in the middle is shaped like the X pentomino so the 5th P won't fit.
Those that can fit with 4 P's: I,L,T,V,X,Y,Z
(pictures on request)
Those that cannot: F,P,N,U,W

Posted by Jer
on 20160304 11:56:12 