SQUARE THE RECTANGLE
Problem, game and general theorem copyright Karl Scherer February 2001

Theorem and Conjecture  List and Diagram  Images of all Tilings

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
11x15
Here is an easy example: a 11 x 15 rectangle tiled in a no-touch 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 and general theorems click here

The General Theorem on Nowhere-Neat Tiling of Rectangles can be found here.
The theorem states that there is always such a tiling as long as the rectangle is sufficiently large.
To be precise, Karl Scherer showed that there is a faultfree nowhere-neat tiling for each R(n, m) with n > 1187 and m > 464. Additionally, each such solution can be chosen to contain the 1x1 square and hence is not an enlargement of a smaller tiling.

Conjecture
Studying all rectangles smaller than 50x50 (see below) we find that
 - a rectangle (< 50x50) which has a faultfree tiling always also has a faultfree tiling which is not an enlargement.
 - all rectangles greater than 20x20 and smaller than 50x50 have a faultfree solution.
This leads to the following conjecture:
It is the author's conjecture that all rectangles and squares of size 22x22 or larger have a faultfree nowhere-neat tiling which is not an enlargement of a smaller tiling.

List of rectangles (< 50x50) with known nowhere-neat, faultfree solutions (images further down, some solutions larger than 50x50 also listed):
------------------------------------------------------------------------------------------------------------------------
11 x 15
12 x 18, (33+15n)
13 x 40, 43, 46, 50, 51, 63
14 x 53, 54, 55, 58, 60
15 x 17, 24, 31, 33, 39-40, 42, 45-47, 49-50, 51, (57+11n), 65, 71, 81, 86
16 x 18, 28, (29+20n), 32-52, 55, 68, 85
17 x 27, 30-32, 34-50, 71, 75, 77, 95, 119
18 x 23-27, 29-50, 52, 56, 63, 70, 71, 72, 75
19 x 21, 25-52, 57, 59, 60, 61, 65, 73, 75, 77, 104
20 x 22-24, 26-50, 53, 54, (56+22n), 61, 63
21 x 23, 26-50, 55-57, 66, 73
22 x 23-50, 53, 56, 59 (all solved up to 50)
23 x 24-50, 55, 106 (all solved up to 50)
24 x 25-50, 52, 56, 70, 77, 85 (all solved up to 50)
25 x 26-50, 57 (all solved up to 50)
26 x 27-50 (all solved up to 50)
27 x 28-50 (all solved up to 50)
28 x 29-50, 57, 58, 86, 168 (all solved up to 50)
29 x 30-51  (all solved up to 50)
30 x 31-50, 52, 72, 74 (all solved up to 50)
31 x 32-50 (all solved up to 50)
32 x 33-50 (all solved up to 50)
33 x 34-50 (all solved up to 50)
34 x 35-50 (all solved up to 50)
35 x 36-50, 73  (all solved up to 50)
36 x 37-50, 57  (all solved up to 50)
37 x 38-50 (all solved up to 50)
38 x 39-50, 58, 77, 100 (all solved up to 50)
39 x 40-50 (all solved up to 50)
40 x 41-50, 54 (all solved up to 50)
41 x 42-50 (all solved up to 50)
42 x 43-50 (all solved up to 50)
43 x 44-50 (all solved up to 50)
44 x 45-50 (all solved up to 50)
45 x 46-50 (all solved up to 50)
46 x 47-50 (all solved up to 50)
47 x 48-50 (all solved up to 50)
48 x 49-50 , 51 (all solved up to 50)
49 x 50 (all solved up to 50)
-------------------------------------------------------------------------------------------
This list contains all rectangles with sidelengths 50 or smaller that have a solution.
All other sizes of
rectangles with sidelengths 50 or smaller have been proven not to have a solution at all!

If possible, a solution should be:
 - fault-free (no straight breaking line running through)
 - not be an enlargement (e.g. a 22x30 tiling can be constructed by enlarging of 11x15 by a factor of two).

Diagram of all solutions:


Green buttons: No-touch tiling exists
Red buttons : Nowhere-neat tiling exists
Light green buttons:  No-touch tiling with fault line exists
Orange buttons:  Nowhere-neat tiling with fault line exists
Cross : no nowhere-neat (and hence no no-touch) tiling exists

Most solutions have been found by Karl Scherer by hand. All rectangles and their solutions can also be found in my Zillions games 'Square The Square', 'Square The Square II', 'Square TheRectangle'  and 'Square-The-Square Solver' which helps you to find new solutions. The related topic of simple perfect and simple imperfect squares and rectangles can be found at  http://www.squaring.net.



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. No-touch solutions are marked by an asterisc *.
or each size of rectangle (smaller than 50x50) we give at least one solution which is fault-free and which is not an enlarged version of another solution.

12x18*

12x(33+15n)

13x40  

13x43  

13x46  

13x50  

13x51  

13x63

14x53

 14x54

14x55

14x58

14x60

15x17

 15x24

15x31

15x33  

15x33  

15x39  

15x40  

15x42  

15x45*

15x46  

15x17  

15x49  

15x50

15x50

 15x51*

15x57

15x(57+11n)

15x65

15x70

  15x71

15x81

15x86

  16x18*

16x28

16x29

16x32  

16x33  

16x34  

  16x35  

16x36 

16x37  

16x38  

16x39  

16x40  

16x41

16x42  

16x43  

16x44

16x44

16x45  

16x45  

16x46  

16x47

16x47

16x47

16x48  

16x48  

16x49  

16x49  

16x50  

16x50  

16x50  

16x50  

16x50  

16x51

16x52

16x55

 16x68

16x85

17x27  

17x30  

 17x31  

17x32  

17x34  

17x34  

17x35  

17x36  

17x37

17x38  

17x39  

17x39  

 17x40  

17x40  

17x41  

17x41  

17x41  

17x42  

17x43  

17x44  

17x45  

17x46

17x47  

17x48  

17x49

17x50  

17x50  

17x50  

17x71

17x77

 17x95

17x105

17x119

18x23  

18x24  

18x25  

18x25  

18x26  

18x27

18x27

 18x29

18x29  

18x29  

18x30*

18x30  

18x30  

18x31  

18x31*

18x31  

18x32  

18x32  

18x32*

18x32  

18x33  

18x34  

18x35  

18x35  

18x35*

18x35  

18x36

18x36

 18x36

18x36

18x37

18x37

18x38

18x38

18x38 

18x39*

18x39*

18x39

18x39

18x40

18x40

18x40

18x41

18x42

18x42

18x42

18x42

18x42

18x42

18x43

18x43

18x43

18x43

18x44

18x44

18x44

18x44

18x44

18x44  

18x45

18x45

18x46*

18x46

18x46

18x46

18x46

18x46

18x46

18x47 

18x47 

18x48  

18x48  

18x48  

18x48  

18x49 

18x49

18x50

18x50

18x52

18x54

18x56

18x63*

18x70

18x71*

18x72 *

18x75

19x21  

19x25

19x26  

 19x27

19x27

19x27  

19x27

  19x28

19x28  

19x29

19x29

19x30

19x30  

19x31

19x31

19x32

19x32

19x33  

19x33  

19x33  

19x33  

19x34

19x35

19x36

19x37 

19x37

19x37

19x38

19x39 

19x39

19x39

19x39  

19x39  

19x40

19x40

19x40  

19x41  

19x41  

19x41 

19x41  

19x41  

19x41  

19x41  

19x42

19x42

19x42

19x42

19x42

19x43

19x43

19x44  

19x44

19x45

19x46

19x46 

19x46

19x46 

19x47

19x48

19x48  

19x49

19x49

19x50

19x50  

19x50  

19x51

19x52

19x57

19x59

19x60

19x60

19x61

19x65

19x73

  19x75*

19x77

19x104 

20x22  

20x23  

20x24

20x24

20x26  

20x27*

20x27  

20x27  

20x28  

20x28  

20x29*

20x30 

20x30

20x30  

20x31  

20x32

20x33*

20x34

20x35

20x35

20x35

20x36

20x36*

20x36  

20x37*

20x37   

20x37  

20x37  

20x38

20x38

20x38  

20x38  

20x38

20x39

20x39

20x40*

20x40*

20x40  

20x40  

 20x40  

20x40  

20x41

20x41

20x41

20x41  

20x42 

20x42

20x42

20x42  

20x42  

20x42  

20x43 

20x43 

20x43 

20x44

20x45*

20x45

  20x45

20x45

20x45

20x45

20x46

20x46 

20x46

20x47

20x48

20x49*

20x49  

20x49

20x50  

20x50  

20x50  

20x50

20x53

20x54

20x56

20x(56+22m)*

20x61

20x63 *

20x63  

21x23  

21x26  

21x27  

21x28  

21x29  

21x30*

21x30  

21x30  

21x31  

21x32  

21x33*

21x33  

21x34  

21x35

21x35*

21x36  

21x37*

21x37  

21x38*

21x38  

21x38  

21x39*

21x39

21x39  

21x39  

21x39  

21x39  

21x40  

21x41*

21x41  

21x41  

21x42  

21x42  

21x43*

21x43  

21x43

21x43  

21x43  

21x43  

21x44

21x44

21x45

21x45

21x45*

21x45  

21x45*

21x46

21x46*

21x46*

21x46*

21x46  

21x46  

21x46  

21x47*

21x47  

21x47

21x47  

21x47  

21x48  

21x49

21x49*

21x49  

21x50  

21x50  

21x50  

21x50  

21x50  

21x50  

21x50  

21x51

21x55*

21x56

21x57*

21x66*

21x73  

20x22  

22x23  

22x24  

22x25  

22x26

22x27*

22x28

22x28*

22x28

22x29

22x29

22x30*

22x31  

22x32

22x33*

22x33

22x34  

22x34  

22x35*

22x36

22x37*

22x37

22x37

22x37*

22x38

22x38

22x39  

 22x40 

22x40  

22x40*

22x40  

22x41

22x41  

22x42

22x42  

22x43

22x43

22x43

22x43

22x43

22x43  

22x44

22x45

22x45

22x45  

22x46*

22x46  

22x47  

22x47  

22x47

22x47

22x48

22x48

22x49

  22x50  

22x50  

22x50  

22x53

22x56*

22x59  

23x24  

23x25  

23x26  

23x27  

23x28  

23x28  

23x29  

23x30

23x30  

23x31*

23x32  

23x32  

23x33*

23x34  

23x34  

23x35  

23x36  

23x37  

23x37  

23x38

23x38

23x39

23x39*

23x39  

23x39  

23x40  

23x41*

23x41 

23x41

23x41  

23x41*

23x41  

23x42  

23x43 

23x43 

23x43  

23x44  

23x44  

23x45*

23x45  

23x45  

23x45  

23x46

23x46  

23x47  

23x47  

23x48

23x49  

23x49  

23x50  

23x50  

23x50  

23x55

23x106*

24x25  

24x26  

24x27

24x28*

24x28

24x28  

24x30  

24x30  

24x29  

24x30  

24x31  

24x32  

24x32  

 24x32  

24x33  

24x34  

24x35  

24x36* (enlarged tiling!)

24x36  

24x36  

24x36  

24x37  

24x37  

24x38  

24x39

24x39

24x40

24x40  

24x41

24x41  

24x42

24x42  

24x42  

24x43*

24x43  

24x44  

24x44*

24x45  

24x45  

24x45  

24x45  

24x46

24x46

24x47  

24x48

24x48

24x49  

24x49  

24x50  

24x50  

24x52

24x56

24x70

24x77

24x77

24x85

25x26

25x27*

25x28  

25x29  

25x30  

25x31  

25x31  

25x32

25x32*

25x32  

25x33*

25x34*

25x34  

25x34  

25x34  

25x35  

25x36  

25x37  

25x38

25x39*

25x39*

25x39

25x39  

25x39  

25x39  

25x40*

25x41*

25x41

25x41*

25x41  

25x42  

25x43  

25x43  

25x44

25x44 

25x44  

25x45

25x45
 
25x45

25x45  

25x46  

25x47

25x47  

25x47  

25x47  

25x48

25x49  

25x49    

25x50

25x57 

26x27  

26x28

26x29

26x30

26x31  

26x32  

26x33*

26x34  

26x34  

26x35  

26x36*

26x36  

26x36  

26x37  

26x37  

26x38*

26x38*

26x38  

26x38  

26x39  

26x40  

26x41

26x41

26x41

26x41  

26x41  

26x42*

26x42  

26x42  

26x43  

26x43  

26x44

26x45

 26x46*

26x46  

26x47

26x47    

26x47  

26x47  

26x47  

26x48 

26x48 

26x49 

26x49 

26x49  

26x50

26x50*

27x28

27x28

27x29  

27x30

27x30

27x31  

27x32

27x32  

27x32  

27x33  

27x34

27x35

27x36  

27x37

27x37  

27x38  

27x38  

27x39*

27x39  

27x39  

27x39  

27x40  

27x40  

27x40

27x41

27x41 

27x41  

27x42  

27x43

27x43

27x44

27x45*

27x45  

27x45 

27x45  

27x46*

27x46  

27x46  

27x47  

27x47  

27x48*

27x48 

27x48  

27x49   

27x50  

28x29  

28x30

28x31  

28x31*

28x32  

28x33*

28x33*

28x33  

28x33    

28x34  

28x35*

28x36*

28x36

28x36

28x37  

28x37  

28x38

28x38  

28x38  

28x39

28x39

28x39  

28x40*

28x40  

28x40  

28x40  

28xs40

28x41  

28x41  

28x42  

28x43  

28x43  

28x44*

28x44  

28x45 

28x45  

28x46*

28x47  

28x48

28x48  

28x49  

28x50

28x57*

28x58*

28x86*

28x168*

29x30  

29x30  

29x31  

29x31  

29x31  

29x32  

29x33*

29x33*

29x34  

29x35  

29x36  

29x36  

29x37

29x37  

29x38  

29x38  

29x39

29x40

29x41

29x41  

29x42  

29x43*

29x43  

29x44

29x45*

29x45  

29x46

29x47 

29x48

29x49

29x50  

29x51*

30x31

30x31

30x31  

30x31  

30x31  

30x32  

30x33

30x34*

30x34  

30x34  

30x35  

30x35  

30x36  

30x36  

30x36  

30x36  

30x37  

30x37*

30x37  

30x37  

30x38*

30x38  

30x38  

30x38  

30x38  

30x38  

30x38  

30x39

30x39

30x39  

30x40  

30x40  

30x41

30x41

30x42

30x42

30x42

30x43  

30x43  

30x43  

30x43  

30x43  

30x44  

30x44  

30x45

30x45

30x45

30x46  

30x46  

30x46  

30x47  

30x47  

30x47*

30x48

30x48

30x49

30x49

30x50*

30x50  

30x50  

30x52

30x72*

30x74

30x74

31x32

31x32

31x32  

31x32  

31x33*

31x33*

31x33  

31x33

31x34  

31x35*

31x35  

31x35  

31x36  

31x36  

31x37  

31x38  

31x38  

31x39  

31x39

31x40

31x40  

31x40  

31x41  

31x42

31x43  

31x43  

31x43   

31x44*

31x44  

31x45*

31x46

31x47*

31x47  

31x47  

31x47  

31x48*

31x49

31x49  

31x50*

32x33*

32x33  

32x33  

32x33*

32x33  

32x34  

32x34  

32x34  

32x34  

32x35  

5 32x36  

32x36  

32x37  

32x37  

32x37  

32x37  

32x38*

32x38  

32x39

32x39

32x40*

32x40  

32x41

32x42  

32x42  

32x42*  

32x43

32x43  

32x43  

32x44

32x44  

32x44  

32x45  

32x45  

32x46  

32x46  

32x46  

32x47  

32x47  

32x48

32x48

32x49

32x49  

32x49  

32x50  

33x34*

33x34*

33x34

33x34*

33x35  

33x35  

33x36

33x36  

33x37  

33x37  

33x38  

33x38*

33x38  

33x38  

33x39

33x40

33x40

33x40  

33x41

33x42

33x42

33x43

33x44

33x44  

33x45

33x45

33x46

33x46

33x47*

33x47  

33x47  

33x48

33x48*

33x49

33x49

33x50

33x50  

34x35  

34x36

34x36  

34x37

34x37  

34x37  

34x38

34x38

34x38  

34x39  

34x40  

34x41  

34x42  

34x42  

34x43

34x44  

34x44  

34x44  

34x44*

34x44  

34x45

34x46

34x47

34x48*

34x48  

34x49  

34x50  

35x36  

35x37*

35x37  

35x37  

35x38  

35x39  

35x39  

35x39

35x40

35x41

35x41

35x41

35x42 

35x42  

35x43  

35x43  

35x44

35x45 

35x46  

35x47

35x47

35x48

35x49 

35x50

35x73*

36x37

36x37*

36x37

36x38

36x39  

36x40*

36x40*

36x41  

36x41  

36x42

36x43  

36x44

36x45

36x46

36x47  

36x47  

36x48*

36x49

36x50

36x50

36x57*

37x38

37x38

37x38  

37x39

37x39  

37x39  

37x40

37x41*

37x41  

37x41