[Puzzler-users] Pentacube puzzles - Octahedral planes

[Stephen L. <ste...@gm...>]

This puzzle is provably impossible:

The puzzle has 19 cubes which are 6-connected.  Simple counting of
orientations will indicate that the following pieces at minimum eat up 15
of the 19 spaces: V1, V2, I, L1, J1, T1, Z, Q, and T eat up at minimum 1
space each and S2, N2 and L3 eat up 2 at minimum.  That leaves only 4 spare
spaces.  Now, considering the following set of 11 pieces: T1, A, S1, N1, Q,
V, L2, J2, L4, J4, T2.  Each piece takes up the fewest 6-connected cells by
being oriented with one of its cells having 5 connectivity.  There are ONLY
6 cells in the model with 5 connectivity.  Hence 5 of these 11 pieces must
contain MORE than its minimum number of 6-connected cells.  Even if this is
just one more, the sum of 5 and 15 is 20 which is greater than the total
number of 6-connected cells.  Hence by the pigeonhole principle, this is
impossible.