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

Theorem List Solutions

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 solution for each R(n, m) with n > 1187 and m > 464.

List of rectangles with known solutions (images further down):
----------------------------------------------------------------------------------------
11 x 15
12 x 18, (33+15n)
13 x 40, 43, 46, 51, 63
14 x 53, 54, 55, 58, 60
15 x 17, 24, 31, 33, 39-40, 45, 49-50, 51, (57+11n), 65, 71, 81, 86
16 x 18, 28, (29+20n), 32, 35, 36, 39, 41, 44-52, 55, 68, 85
17 x 30, 32, 34, 36-41, 44, 46, 49-50, 71, 75, 77, 95, 119
18 x 23-25, 27, 29-32, 34-50, 52, 56, 63, 70, 71, 72, 75
19 x 21, 25-52, 57, 59, 60, 61, 65, 73, 75, 77, 104
20 x 24, 26-50, 53, 54, (56+22n), 61, 63
21 x 23, 26, 29-50, 55-57, 66, 73
22 x 23-24, 26-50, 53, 56, 59
23 x 28-50, 55, 106
24 x 27-28, 30, 32-50, 52, 56, 70, 77, 85
25 x 27, 30-34, 36, 38-50, 57
26 x 28-31, 33-38, 40-50
27 x 28-30, 32, 35-41, 43, 45-50
28 x 30-33, 35-50, 57, 58, 86, 168
29 x 30-34, 36-51,
30 x 31-50, 52, 72, 74 (all solved up to 50)
31 x 32-40, 42-50
32 x 33-50 (all solved up to 50)
33 x 34-50 (all solved up to 50)
34 x 35-40, 42-50
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 are known to have a solution.
Can you find a few more?
Solutions should:
 - be fault-free (no straight breaking line running through)
 - not be an enlargements (e.g. 22x30 can be constructed by enlarging 11x15 by a factor of two).

Most of the solutions have been found by Karl Scherer; some have been provided by Joseph DeVincentis and Alfred Pfeiffer. All rectangles and their solutions can also be found in my Zillions game 'Square The Square II' 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 *.
If possible, solutions should also be fault-free and should not be an enlarged version of another solution.

12x18*

12x(33+15n)

13x40

13x43

13x46

13x51

13x63

14x53

 14x54

14x55

14x58

14x60

15x17

 15x24

15x31

15x33  

15x33  

15x39  

15x40

15x45*

15x49  

15x50

15x50

 15x51*

15x57

15x(57+11n)

15x65

15x70

  15x71

15x81

15x86

  16x18*

16x28

16x29

16x32  

  16x35

16x36 

16x39  

16x41

16x44

16x44

16x45  

16x45  

16x46  

16x47

16x47

16x47

16x48  

16x48  

16x49  

16x49  

16x50  

16x50  

16x50  

16x50  

16x50  

16x51

16x52

16x55

 16x68

16x85

17x30  

17x32  

17x34  

17x36  

17x37

17x38  

17x39  

17x39  

 17x40  

17x40  

17x41  

17x41  

17x41  

17x44  

17x46

17x49

17x50  

17x50  

17x50  

17x71

17x77

 17x95

17x105

17x119

18x23  

18x24  

18x25  

18x25  

18x27

18x27

 18x29

18x29  

18x29  

18x30*

18x30  

18x30  

18x31

18x31*

18x31

18x32

18x32

18x32*

18x32

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 

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  

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  

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  

22x23  

22x24  

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

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*

24x27

24x28*

24x28

24x28  

24x30  

24x30  

24x30  

24x32  

24x32  

 24x32  

24x33  

24x34  

24x35  

24x36*

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

25x27*

25x30  

25x31  

25x31  

25x32

25x32*

25x32  

25x33*

25x34*

25x34  

25x34  

25x36  

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 

26x28

26x29

26x30

26x31  

26x33*

26x34  

26x34  

26x35  

26x36*

26x36  

26x36  

26x37  

26x37  

26x38*

26x38*

26x38  

26x38  

26x40  

26x41

26x41

26x41

26x41  

26x41  

26x42*

26x42  

26x42  

26x43  

26x43  

26x44

26x45

 26x46*

26x46  

26x47