Pentomino Fuzion Puzzles

# Pentomino Fuzion Puzzles

The puzzle piece shown here is the "Z" pentomino. Pentominoes are named after dominoes but are made from five squares. Polyomino is the general term for all the different classes: two for dominoes, six for hexominoes, etc. Each piece is formed by joining equal sized squares together along complete edges not at corner points. The study of polyomino arrangements is an exact mathematical science (with limited practical applications ;-) For example, there are exactly 12 different pentominoes formed by five squares. If you are skeptical, look away from the graphics on this site and try drawing every different pentomino you can think of. Pieces that can be made to look the same by rotation or flipping-over are considered the same. It may take awhile to discover them all but you will see there are exactly 12 and no more. Okay, now try to draw all 35 hexominoes made from six squares or 108 heptominoes made from seven squares! Answers are in Polyomino Statistics , but that spoils the fun.

## Pentomino Puzzles

The challenge is to fit all the pieces together like a puzzle. This may seem easy considering there are thousands of solutions, but it is quite tough to find even one without computer help. Since there are five squares per pentomino and twelve different pieces, they should fit on boards with exactly sixty squares. Sixty is very factorable, making it possible to solve rectangular 6X10, 5X12, 4X15, and 3X20 boards. It is well known that the 6X10 board has exactly 2339 different solutions while the 3X20 board has only 2. Many other interesting board shapes made from sixty squares have been proposed and solved. Each solved board can be flipped-over on the long side, the short side, and both ways combined making it appear there are four different solutions. These rotations and reflections are not considered unique solutions. However, there are 9356 different ways to solve a new 6X10 pentomino board. Can you find solutions to these challenging puzzles?

## Hexomino Puzzles

There are 35 ways to fit together 6 squares so hexominoes fit on boards with exactly 210 squares. There are no known solutions to regular rectangular hexomino boards (the one below has "holes") or any other "even checkerboard" pattern. The theory in Golomb's book shows that rectangular boards are impossible because 24 hexominoes have an "odd" and 11 have an "even" checkerboard pattern. This gets a bit complicated, but here's how I understand the theory. Imagine placing each hexomino on a checkerboard. It can be seen that 24 hexominoes have a pattern with 3 white and 3 black squares, these "cancel" each other out. The remaining 11 "even" pieces have 4 white and 2 black squares (or vice versa). So, since 10 of the 11 can "cancel" each other (a 4 white-2 black cancels a 2 white-4 black), the last piece will have an extra white or black. Therefore, boards with 106 white and 104 black squares (or vice versa) are solvable but since rectangular boards have 105 whites and 105 blacks they can not be solved. By similar reasoning, boards with 108 whites and 102 blacks for example are also solvable. Generally, boards with symmetrical hole patterns are not solvable. I wonder how long they tried to solve the boards before deciding it would be easier to prove it couldn't be done.

When designing a new board check for a valid checkerboard pattern, unless you are trying to find an exception to Golomb's rule. Good luck! It is very difficult to find hexomino solutions, even using Fuzion's warp speed search algorithm. Here are several solved hexomino boards. Try carefully fitting together approximately 20 of the 35 pieces before launching an Auto search and rearrange if the near-miss counters don't pile-up after about 15 minutes. Using the fastest Fuzion search algorithm on present day PC computers, it takes months to try all possible arrangements to solve a hexomino board compared to minutes for pentominoes.

# Fuzion

Fusion makes sunshine by combining atoms. Fuzion is a shareware program combining polyomino shapes in a variety of colorful puzzle games. You can play with either the 12 familiar pentominoes or 35 spectacular hexominoes. Fuzion's automatic solve mode features an amazing visual algorithm that exhaustively locates all possible solutions. Start an automatic search after manually placing as many pieces as you can. Interrupt, modify, and restart the search anytime. The program displays a library of solutions which can be retrieved to the manual board as starting points. The library grows as each new solution you find is saved. Most of the examples and statistics on this site were generated and confirmed on Fuzion. Fuzion brings the power of PC computers to this fascinating diversion that has intrigued puzzle enthusiasts for years.     Download Fuzion shareware v2.8.NOV2001 (95 kb)

## Fitz

Fuzion includes several polyomino based games. This newest pentomino game stretches your visualization powers in a race to fit as many pieces as possible. Five pieces are randomly landed, then you continue landing any piece that keeps a complete solution possible. When those areas are exhausted, press the number key of any remaining piece that fits anywhere. You receive extra Ponder Points for quick thinking. Some pieces will only fit in one place and should be pressed first, then fit the pieces with multiple landing areas to increase your score. Once at least twelve rounds are completed, your score is eligible for the best on the web Hall of Fame.

## KnoZone

The object of KnoZone is to place the pentomino you select so your opponent will be unable to place his final piece. Opponents (human or the computer Al Grim) alternate turns selecting any one of the remaining pentominoes. There's a lot of flexibility with the first few moves, but plan ahead or there will be no zone for your last turn. If you can not place any of the remaining pieces (or you don't see a move) the round is conceded. The round winner receives one point for each unplaced piece. The player starting the next round is randomly selected. Careful, once a selected pentomino is placed on the board it must be used and kept pieces can not be moved again.

## Zurvivors

Zurvivors is a pentomino game on Fuzion where you place six of the twelve pieces in an arrangement that hopefully will be part of a solved puzzle. Your six pieces are randomly selected at the start of a turn and can not be changed. Make your best arrangement then press Auto to start a computer solution search using your base. Make sure there are no overlaps, out of bounds, or trapped blank spaces not divisible by 5 or a solution will be impossible. First all six pieces are checked for being part of a solution. If one isn't found, your sixth piece moves off into the search pool and your remaining five pieces are checked. This continues until a solution is found. You receive one point for each piece left where you placed it. Once at least six rounds are completed, your score is eligible for the best on the web Hall of Fame.

Send E-mail about pentominoes or Fuzion

Click on Fuzion Home to transfer to the most current Fuzion site.