Jan 29, 2002 From: Richard Guy I got interested in the number of ways of packing calissons (diamonds, rhombuses) in a hexagon. I laboriously calculated 1, 2, 20, 980 and sent this to `lookup', only to find that this was well known to those who well know it as A008793. But I'm also interested in the number of ways, not counting rotations and reflexions as different. n=0. 1 way, by a usual convention. n=1. 1 way (rotate thru pi/6 to get the other) n=2. 6 ways: ________ ________ ________ / /\ \ /\ \ \ /\ \ \ /___/ \___\ / \___\___\ / \___\___\ /\ \ / /\ /\ /\ \ \ /\ / /\ \ / \___\/___/ \ / \/ \___\___\ / \/___/ \___\ \ / /\ \ / \ /\ / / / \ /\ \ / / \/___/ \___\/ \/ \/___/___/ \/ \___\/___/ \ \ / / \ / / / \ / / / \___\/___/ \/___/___/ \/___/___/ 1 of these 2 of each of these ________ ________ ________ / /\ \ /\ \ \ / /\ \ /___/ \___\ / \___\___\ /___/ \___\ / /\ /\ \ /\ / /\ \ /\ \ / /\ /___/ \/ \___\ / \/___/ \___\ / \___\/___/ \ \ \ /\ / / \ / /\ / / \ /\ \ \ / \___\/ \/___/ \/___/ \/___/ \/ \___\___\/ \ \ / / \ \ / / \ / / / \___\/___/ \___\/___/ \/___/___/ 3 of these 6 of each of these makes 20 in all. n=3. 113 ways. I won't draw them all, but here is a check: ____________ 8 x 2 = 16 / /\ \ \ 1 x 4 = 4 /___/ \___\___\ 48 x 6 = 288 / /\ / / /\ 56 x 12 = 672 /___/ \/___/___/ \ ---- ----- /\ \ / /\ \ /\ 113 980 / \___\/___/ \___\/ \ \ /\ \ \ /\ \ / \/ \___\___\/ \___\/ \ / / /\ / / \/___/___/ \/___/ \ \ \ / / \___\___\/___/ Here's the unique one. ################################################################### Postscript Date: Thu, 07 Feb 2002 10:23:10 +0000 From: Don Reble For n=4, 20174 ways. Similarly, 2 x 1 = 2 33 x 2 = 66 18 x 3 = 54 16 x 4 = 64 1433 x 6 = 8598 18672 x 12 = 224064 ----- ------ 20174 232848 Here are both fully-symmetric ones. ________________ / / /\ \ \ /___/___/ \___\___\ / / /\ /\ \ \ /___/___/ \/ \___\___\ /\ \ \ /\ / / /\ / \___\___\/ \/___/___/ \ /\ /\ \ \ / / /\ /\ / \/ \___\___\/___/___/ \/ \ \ /\ / / /\ \ \ /\ / \/ \/___/___/ \___\___\/ \/ \ / / /\ /\ \ \ / \/___/___/ \/ \___\___\/ \ \ \ /\ / / / \ __\___\/ \/___/___/ \ \ \ / / / \___\___\/___/___/ ________________ / / /\ \ \ /___/___/ \___\___\ / /\ \ / /\ \ /___/ \___\/___/ \___\ /\ \ / /\ \ / /\ / \___\/___/ \___\/___/ \ /\ / /\ \ / /\ \ /\ / \/___/ \___\/___/ \___\/ \ \ /\ \ / /\ \ / /\ / \/ \___\/___/ \___\/___/ \/ \ / /\ \ / /\ \ / \/___/ \___\/___/ \___\/ \ \ / /\ \ / / \___\/___/ \___\/___/ \ \ \ / / / \___\___\/___/___/