Square The Square

SQUARE THE SQUARE
Problem, game and general theorem copyright Karl Scherer February 2001.
Last update: February 2006

Definition of a "nowhere-neat tiling" and a "no-touch tiling"
You are given a (n x n) square or a (n x m) rectangle that you have to tile with squares in such a way that no two tiles have a full side in common. Such a tiling is called "nowhere-neat".
(See also my books "NUTTS And Other Crackers" and "New Mosaics".)
If tiles of same size are not allowed to share any part of a side, the tiling may be called "no-touch".

Example

11x11
Here is an easy example: a 11 x 11 square tiled in a nowhere-neat way.

Problems
Now try the find nowhere-neat tilings or no-touch tilings for the following squares and rectangles:
(Scroll down to see some of the the solutions.)

Squares with sidelength: n = 16, 18, 19, 20 and all n > 21. Solutions see below.
Rectangle problems and solutions: click here.

(See also Joseph Devincentis' page. He is the solver of at least one of the squares of size 16, 19, 20, 23, 27, 32, 33, 35, 37, 38, 39, 40, 45, 47, 49, 71 amongst the 90 diagrams displayed here).
The Solution to square 71x71 is especially interesting in that it does not contain a 1x1 square tile!

In March 2001 Karl Scherer proved the following two theorems:
(Note that no-touch tilings are always nowhere-neat)

General Theorem for nowhere-neat tilings of a square with squares.
A square has a nowhere-neat tiling if its sidelength is 11, 16, 18, 19, 20 or greater than 21.
The tiling can always be chosen to contain the unit square 1x1 as a tile.
Additionally the tilings can be chosen to be faultfree except for the case n = 22.

General Theorem for no-touch tilings of a square with squares.
A square has a no-touch tiling if its sidelength is 16, 23 or greater than 24 (*).
The tiling can always be chosen to be faultfree and also contain the unit square 1x1 as a tile.

My paper showing the proof has been published in the Journal Of Recreational Mathematics 2003-2004, Vol 32(1), pages 1-13.
(*) Patrick Hamlyn later found solutions for s=18, 22, 24 as well, see below or follow this link.
To see a list of references for further reading on related topics click here.

Game
The games "Square The Square", "Square The Square II", "Square The Rectangle" are available as a Zillions computer game, free for you to download from my Zillions game page. All the solutions and images presented here have been created using these Zillions games as drawing tools. I also claim copyright for the idea to use this tiling problem as a real board game with square tiles.

Solutions:
Note that several cases have more than one solution.
Solutions which are enlargements of other solutions. Also, solutions with faultlines have been omitted (except for n=22)..
No-touch solutions are marked by an asterisc *.

16x16*