On 8/28/2013 8:19 PM, David Wilson wrote:
Consider a 3-D tile made of 5 cubies arranged like a P-pentomino, 1 cubie thick.
You pack these into a 3x3 cube (together they fill all but 2 cubies).
Suppose one of the unpacked cubies is the center cubie.
Can the remaining unpacked cubie be at a vertex of the 3x3? An edge? A face center?
A pleasant little puzzle to solve at the end of a long day. (BTW, anyone wondering what Allan meant can find the Bill Cosby routine here: http://www.youtube.com/watch?v=bputeFGXEjA ) Spoiler below: A P-pentomino is planar, so each tile lies within a one-cubie-thick slice of the cube. Since the center is unused, such a slice consists of the nine cubies on one face of the cube. The face center cubies each belong to only one face, so the five tiles can only use five of them. Thus one face center must remain unused. It remains to show that such a packing is possible. Here's one, layer-by-layer: 2 2 1 3 1 1 3 1 1 2 2 2 3 x 4 3 x 4 5 5 4 5 5 4 3 5 4 -- Fred W. Helenius fredh@ix.netcom.com