The figure below resembles a staircase, so call it a staircase polyomino. (This instance is size 5)
X
XX
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XXXX
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Give a staircase polyomino unit thickness to form a solid. It is trivial to pack n of the sizen staircase polyominos into a nxnxn cube.
Prove or provide a counterexample to the statement: It is impossible to pack n+1 of the sizen polyominos into a nxnxn cube.
n=1
The staircase is just a 1x1x1 cube and you clearly can't pack 2 of them in a 1x1x1 cube.
n=2
The volume of the 3 staircases is 9, but the 2x2x2 cube has volume 8.
n=3 probably won't work either.
I suppose the statement should have ended with "for any n."
Then one can try to find a value for which it does work.

Posted by Jer
on 20170310 11:58:09 