Squarematch Facts & Figures
In this game you have a board and some colored tiles.
Each tile consists of four squares, colored separately.
The twocolor version has these 6 tiles.


and the threecolor version 24,
including these 6, one of each type:

The full 70tile fourcolor version is a monster, but we
examine some subsets.
Counting numbers of tiles
The object of the game is to place the tiles on the interior
part of the board, rotating them as necessary, so that wherever
two of them meet, the colors match,
as in the examples above. With a small set such as those
shown, it's not hard. As the set gets larger, it is much more
of a challenge. You can try it with the Play applet.
Kadon Enterprises offers the
game as a physical set under the name
Multimatch^{®} II.
When we speak of solutions to these games, we
ignore rotation, reflection, and interchange of colors. All these
forms of symmetry simply multiply the number of
"interesting" solutions.
There are several 'levels' at which the game can be played:

The beginner level game uses two colors.
Forming a 2 x 3 array can only be done in one essentially different way,
but the 1 x 6 has a total of five.
The solution shown above is cylindrical, so called because the two
side edges match and they could be wrapped around a cylinder rather
than appearing in the plane.
For more details see the 2color Analysis page.

In the intermediate level game (4 x 6 with three colors)
there are 206 different ways to place the solid (1color) tiles,
and all but one have solutions. You may be surprised to learn
which one it is! (See below.) The numbers of solutions for
various placements of the solid tiles range from 3 to 128. The
total number is 12334. Of these solutions, 180 are
cylinders.
A variation of that is a 3x8 board. With fewer interior interactions
(or more border) there is more flexibility and there are more solutions.
There are 230 ways to place the solids, with numbers of solutions ranging from
25 to 303, and a total of 36927.

In the expert level game (5 x 5, three colors, hole in
center) there are 121 different ways of placing the
solid (1color) tiles and each has solutions, ranging in number
from 13 to 302. The total number is 17281.
Easy and Hard Configurations
On the 5x5 board, all placements of the solid tiles lead to
solutions.
The easiest and hardest configurations for the solids are:
302 solutions






13 solutions

However, on the 4x6 board, there are no solutions with all
three solids in corners.
This is a surprising result.
(Well, it is until you think about it.
The four colors in the corners of the rectangle are not paired with adjacent
colors. They are paired with each other. If three of the four corners are
occupied by three the different colors, the fourth can only pair one of them.
The other two would be left as "odd men out". But each color comes in an
even number, so that cannot be.)
The easiest and hardest configurations for the solids are:
128 solutions




3 solutions




NO solutions !

For the 3x8, the hardest and easiest configurations are shown.
Like the 4x6, there are no "all solids in corners" solutions.
Note also the similarity of the configurations for the highest number
of solutions with that of the 4x6.
25 solutions




303 solutions

To see some complete solutions for these configurations, view
the Special Solutions page.