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Happy Pentominos

 
pentomino_pieces
Last updated Wed Nov 4 1998 Last updated Sun Nov 1 1998

Four Perfect Rectangles

The 12 pentominoes occupy 12x5 = 60 squares and by a happy coincidence, 60 is also a very nice number. It has plenty of divisors: 2, 3, 4, 5, 6, 12, 15, 30. The rectangles 3x20, 4x15, 5x12, 6x10 are perfect and were naturally the first to be studied. They stand out in red in the general catalog. The 2x30 rectangle is impossible, right?

Immortal And Mortal Rectangles

The imperfect rectangles (for example 6x11 = 66) contain extra pieces. And moving these piece around leads to many inequivalent boards.

There will be boards with the least number of solutions, and boards with the most. In cases where the least number is actually 0, then the next smallest number becomes interesting. A board with no solution is dead. Rectangles that have dead boards are mortal. Rectangles that have no dead boards are immortal. A board with the least non-zero number of solutions is lucky (to still have a solution!) and the least number of solutions is the lucky number. A board with the most number of solutions is happy and this highest number is the happy number.

Well, it turns out that with just 4 moveable pieces any imperfect rectangle has at least one dead board. Can YOU prove that? Big hint: look at the 8x8 board called "the crab". Why the name? That's the proof! So any rectangle with WxH >= 64 is mortal.

Fine, you say, but that still leaves boards with 1, 2 and 3 moveable pieces that are potentially immortal. Only rectangles with 63 squares need to be considered. Do you agree? Now 63 can be made in only 2 ways: 3x21 and 9x7. For the 3x21 rectangle I have already found a dead board. So the only possible immortal rectangle left is 7x9. And now we know enough to state our 3 questions:

Three Questions

Q1: Is the 7x9 rectangle immortal?
It certainly looks that way but it will be harder to prove. In the meantime,
determine the lucky number for the 7x9 rectangle and find a lucky 7x9 board.
Q2: Determine the lucky number for the mortal rectangles and find a lucky board for each one.
Q3: Determine the happy number for all the imperfect rectangles and find a happy board for each one.

I'll keep a list of people who find a best answer to Q1, and I'll show that number and the list, but not the board!

A last, and easy, observation is that there are only a finite number of WxH dimensions for which there exist connected boards with solutions. One could also investigate those limits. But here we'll just note that they provides a natural size limit to the catalog.