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Happy Pentominos

 
pentomino_pieces
Last updated Wed Nov 4 1998

The F Pentomino Is A Methuselah

In John Conway's Game of Life, the F pentomino has a secure reputation as a methuselah. If you don't know about Conway's Game of Life, it is well worth investigating. Many sites provide Java applets. In particular, Alan Hensel. There you can open RPENTO.LIF and play it. The GoL people keep on calling it the R pentomino... what do they know!

Counting Poly-ominoes

There is one domino and 12 pentominoes. In trying to count the N-ominoes, nobody has been able to go much beyond the 24-ominoes. This series of numbers is a documented Integer Series. To quote from another site :
The number of n cell polyominoes (that allow holes) for n = 1,2,...,15, is 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238951, 901971, 3426576.

This is sequence A000105(M1425) in Sloane's database of integer sequences.

The number of n cell polyominoes (without holes) for n = 1,2,...,15, is 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001211. This is sequence A000104(M1424) in Sloane's database of integer sequences.

The program used is due to John Boyer, and is based on a paper of Redelmeier (Discrete Math. 36 (1981) 191-203).

And... connections, connections, Neil Sloane and I work in the same building, recently named Shannon Lab! And here is a link linking Golomb, Sloane and Shannon!

Lattice Animals

In 2D percolation theory, N-ominoes are refered to as lattice animals. Now that is cute! This link also refers to work not only by some of the above suspects, J. Conway and S. W. Golomb, but also by R. L. Rivest of MIT, the R in the famous RSA encryption algorithm. These are only the most famous names, of course. The point is, if they can spend a few hours with pentominoes, so can you! (and me!)

Now, from percolation theory to java... hmm, I don't smell a real connection there!