The following 3x4 rectangle can be cut into pieces along grid lines, so that each piece has exactly three neighbors:
Problem: Find the smallest rectangle on the integer grid that can be cut into pieces along grid lines, so that each piece has exactly five neighbors.
Pieces are neighbors, if their boundaries touch; touching in a corner doesn't count.
Answer
How's this?
I got this by square-izing the icosahedron graph:
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