205 was also the best I could do. For diagrams of the packing, see http://groups.google.com/group/rec.puzzles/msg/6f9aadb1e28f8d81. David ----- Original Message ----- From: "Richard Schroeppel" <rcs@cs.arizona.edu> To: <math-fun@mailman.xmission.com> Cc: <rcs@cs.arizona.edu> Sent: Wednesday, October 24, 2007 02:35 Subject: [math-fun] packing 7x11x13 box puzzle
David Cantrell asks ...
How many boxes, each measuring 7 inches x 11 inches x 13 inches, can be packed into a crate measuring 4 feet x 5 feet x 6 feet?
I'll start the bidding at 205.
The trivial upper bound is 48x60x72/7x11x13 = 207 boxes, + 153 in^3 left over.
I get 205 with a mundane packing: Four 11x13 rectangles can be packed into a 24x24 square, with a 2x2 hole in the middle. This tiles a 48x72 layer, 7" thick, with 24 boxes, and a wastage of 168 in^3 per layer. Regard the 60" dimension as vertical, and use the lower 49" to pack seven layers. This gives 168 boxes, wastage 1176 in^3, and a leftover slice 11" thick by 48x72. The slice can hold a maximum of 37 boxes (11x48x72/7x11x13 = 37 + 89/91). One way is to pack 7x13 rectangles into a 48x72 box. Cut out a 35x65 piece (= 25 rectangles), leaving a 13x72 stripe (= 10 ractangles) and a 35x7 stripe (= 2 rectangles). This gives 37 boxes in the top 11" slice, with a wastage of 979 in^3. Totalling up, 205 boxes, wastage 2155 in^3.
Do I hear 206?
Rich
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