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
, but that spoils the fun.
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
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
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
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)
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.
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 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.