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POLYFORMS

Livio Zucca is now selling the sexehexes at his site.  He has a contest on his page on polyforms.
William Waite has many interesting polyform puzzles he cuts from wood, in particular the Interlace Circle and Mixed Angle puzzles.
Mira Vicher has enumerated the polystrips.  Both the 13 hexastrips and 30 heptastrips make rectangles.
Clifford Pickover has put together a nice puzzle at alientiles.com.  There are some group theory aspects to this which should make complete analysis easy.
Andrew Clarke enumerated all possible ways to remove half a triangle from the pentiamonds, with restrictions.  It turns out that the truncated pyramid has a unique solution.  Can you find it?

Mark Michell once needed to make a set of pentominoes with a simple saw.  What is the smallest piece of wood he could have used?  He has put together a page on the solution.  One obvious new question: what is the smallest piece of wood need to make the 12 hexiamonds?

A list of the 1494 solutions for fitting the 14 tetratans into a certain shape is presented here.

http://www.ics.uci.edu/~eppstein/junkyard/polyomino.html has some of the links I'm missing.

Here's how to pack the octominoes:

A page on polyform tilings is here.

Patrick Hamlyn has come up with an insanely difficult puzzle, which I might produce with some sort of prize. I have Patrick's solution for this puzzle locked in my desk.  If I make cubic hexomino sets, they'll come in a cube with the pieces in three colors.  A similar, simpler puzzle with fifteen pieces is the threecolor puzzle (also with assistance from Patrick Hamlyn).  Rearrange the pieces of the image to form another rectangle with no two pieces of the same color touching each other.  No-one has solve that one, so I offer a hint: the whole solution was constructed around X.

Hamlyn's Challenge -- make a 12x18 rectangle so that no two pieces of the same color touch each other.

Has anyone explored the polyforms made with solidly connected coins?  The following diagram shows one way that coins can be stacked.  Imagine another layer with 3 coins on top.  How many different ways can 4 coins be glued together with face to face contact.  Can any interesting figures be made with complete or partial sets of tetracoins?

Ali Muñiz sent me a letter asking about 2-colored tetrominoes.  It turns out this is very similar to Fill-Agree by Kadon Enterprises.  That set shows all twenty ways to put two holes into a tetromino.  If you also look at 0, 1, 3, and 4 holes, it turns out there are 54 possible pieces.  Is it possible to make 6 simultaneous 6x6 squares with 18 holes going all the way through? Ali Muniz, by the way, has  a page devoted to polyomino covering problems.

Thinking more on Fill-Agree by Kadon Enterprises, I asked Michael Reid about all the ways one could drill a hole through a solid L tromino. Start with the tromino, pick any two faces, then connect them with a tunnel. Should these be tunneled polyominoes?  Spelunkinoes?  Anyway, he found 34 of them eliminating rotations and reflections; and 49 eliminating rotations only.  I'm in the process of checking these numbers.  Parity seems to rule out a single closed path, but an open path seems possible.  Can the 49 holey L-trominoes pack into a 3x7x7 box with a single path?  Can 48 of them pack into a 4x6x6 box with a single path?  The set could be put together by drilling 147 cubes and gluing them together.

He found 9 solid tunneled dominoes (8 if reflections are eliminated), and managed to pack them into a 2x3x3 block with a continuous path.  It's a nice simple puzzle.  Here is his solution.

Henri Picciotto has used polyforms in the classroom for ahile now.  He's built a page for them here.  He's also one of the authors of an upcoming Zome book, and occasionally makes fiendish cryptic crosswords.  Henri asked me to cut sets of the Polyarcs.  I likely will, soon.

A polyomino is a figure made of solidly connected squares.  There is 1 omino (a square), 1 domino, 2 triominoes, 5 tetrominoes, 12 pentominoes, 35 hexominoes, 108 heptominoes, 369 octominoes.  Kadon Enterprises sells all of these sets.  Diagrams of everything are available at their site.

From from Geometry Catalog from Nasco (1-800-558-9595), you can get TB16756T -- 1000 1cm cubes (\$20).  Perfect for exploring cubic packing problems.

Some nice polyomino sites include the following....

Gerard's Universal Polyomino Solver has a nice Java applet for solving Pentomino problems.

The Combinatorial Object Server contains a program that will list n-ominoes, and also has a pentomino solving program.

The Pentomino Fuzion Page has programs for pentominoes and hexominoes.

Miroslav Vicher's site has several pages of polyform solutions.

Michael Keller's World Game Review has several issues devoted to polyforms

Rodolfo Kurchan's Puzzle Fun is a magazine devoted to polyforms.

The 26 tridominoes, and a 12 domino figure which contains them all

There are 26 ways to put 3 dominoes together.  Dick Saunders Jr calls them (CAT, CRANKSHAFT, PANCAKES, MAN, LEDGE, HOCKEY, PANTS, CALF, TRIP, WAITER, THUMBTACK, SEAT, TOENAIL, TOUCH TOES, BASEBALL, PROPELLER, HATCHET, MAILBOX, ACHILLES, CROSSING GUARD, CARROT COINS, RECORD PLAYER, and SIT).  Rodolfo Kurchan has found a 12-domino solution that contains all 26 combinations.  Is there a good method for solving problems of this type?  Write me.  In the meantime, I've launched a \$500 contest for the Chaos Tiles.

Related to Polyominoes are the Polyiamonds.