Object: Fill the board with smaller square tiles.
You place a square tile by clicking the top left and the bottom right corner
position of the square where the tile should be placed.
First click for the top left corner, then for the bottom right corner.
The largest available tile is a square of size 18x18.
If you want to place a 1x1 square, click the same position twice.
You may delete an existing square tile by clicking its top left corner
and then its bottom right corner.
You win if you can completely tile the board with smaller squares.
There is an additional restriction, however, depending on the variant:
First set of variants: No four squares are allowed to meet at any point.
We will call this a 'cross-free' tiling.
We present the problem for the sizes 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17 and 21,
because solutions for all other sizes can be found in the second set of variants.
Second set of variants: (More difficult!) No two tiles may have a full side in common.
This is called a NOWHERE-NEAT tiling.
This problem has a solution for any square nxn, where n = 11, 16, 18, 19, 20
or n>21. We even can ask that one of the tiles is a 1x1 square.
If the square tiles of same size are not adjacent at all (apart from touching at a corner),
we call the tiling NO-TOUCH tiling. NO-TOUCH implies NOWHERE-NEAT.
There are no-touch tilings for square sizes n = 16, 18, and all n>21.
For experts: try to use as few tiles as possible for each challenge.
The solutions given are the ones with the fewest tiles,
Choose menu Help/Solution to see one solution per variant.
Some of the solutions have been found by Joseph DeVincentis.
The general nowhere-neat problem was solved by the author in February 2001.
Also a similar theorem on 'no-touch tilings' was found by the author at the
same time (see attached file SqTSq_proof.zip for details).
These theorems have now been published in the Journal of Recreational Mathematics
2003-2004, Vol 32(1), pages 1-13.
Patrick Hamlyn later (in 2005) found no-touch tilings for n=18, 22, and 24.
In 2005 the author also proved similar theorems for squaring rectangles.
Related games : 'Square The Square II', 'Square The Rectangle', 'Square The Square Solver'.
More freeware as well as real puzzles and games at my home page http://karl.kiwi.gen.nz.
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