SQUARE THE SQUARE
Problem, game and general theorem copyright Karl Scherer February 2001.
Last update: Nov 2011

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.



Associated Zillions games: 
"Square The Square", "Square The Square II", "Square The Rectangle"
Associated Wolfram Demonstrations:
Nowhere-Neat Squaring the Square 

Solutions :
Note that most large squares have more than one solution.
Solutions which are enlargements of other solutions and solutions with fault lines have been ommitted.
A star (*) denotes a no-touch solution.

16x16*

18x18

18x18*

19x19

19x19

20x20

22x22*

22x22

22x22

22x22

23x23*

23x23

24x24

24x24*

25x25

25x25

25x25*

  26x26*

26x26

26x26

27x27* 

27x27

27x27*

27x27

28x28*

28x28

29x29

29x29

29x29

29x29*

30x30

30x30

  30x30*

30x30*

30x30*

30x30

31x31*

31x31

31x31*

32x32

32x32 *

32x32

32x32

32x32

32x32

32x32*

33x33

33x33 *

34x34 *

34x34

34x34

35x35

35x35

35x35 *

35x35

35x35

  36x36 *

36x36

36x36

37x37 *

37x37

37x37

37x37

38x38

38x38 *

38x38

39x39 *

39x39 *

39x39 *

39x39 *

39x39

40x40 *

40x40

41x41

41x41 *

42x42 *

42x42 *

43x43 *

44x44 *

45x45 *

45x45 *

45x45 *

45x45 *

45x45 *

45x45 *

45x45

46x46 *

46x46 *

46x46 *

46x46 *

47x47 *

47x47 *

48x48 *

48x48

49x49 *

49x49

71x71 *


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