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Happy Pentominos

 
pentomino_pieces
Last updated Sun Oct 25 1998

My Twenty Something Years Of Pentominoes

My current foray into pentominoes is because I wanted to do some Java, and had this old Quick Basic code (1992) lying around, waiting for a new lease on life. I had written the QB code to get lively color pictures of board solutions. An earlier C program (1984) had dutifully found the 2339 solutions of the 10x6 board but produced only black and white pictures. The C code itself was a rewrite of a BASIC program written for an Apple ][ a year or so before. That Apple ran for 2 weeks on the 10x6 board, beeping and scratching solutions around the clock! And before that I remember playing with pentomino pieces in the 70's (and 60's!) and filling a notebook with solutions. Where that game came from, I never enquired, but my parents probably had something to do with it...

Life, Liberty and the Pursuit of Happiness!

This time what I'm focusing on is the number of solutions that exist for any given board. Somewhere out there is a board that has the MOST tilings of any board, anywhere, and I want to find it! The more solutions, the happier the pentominos are on that board, naturally.

I first found (Oct 11 '98) a board with more than 13000 solutions. And I realized that for 20 years I'd implicitely thought 2339 was tops! A day later I found one with more than 14000 solutions, and then, my current champion with 16720 solutions! To see what I'm talking about, run the applet and select the board named "7x9 happy!" Run it and watch those solutions pepper the screen like pop corn!

So the goal is to build a catalog: for given rectangular dimensions, find the highest number of solutions that are possible on a board constrained by, an reaching, those dimensions. The board shape yielding the highest count is the happy board for those dimensions! I've given it a crack (using the applet, of course) and put a couple of what I think are happy boards on the applet menu. But I didn't prove anything and I didn't do an exhaustive search, so maybe YOU can find happier boards yet!

To assist you, the applet now has a board editor. The mouse pointer wil indicate the pieces than can be moved by dragging (oct 25) and the board itself can be resized by dragging two of the sides (nov 1).