Square The Rectangle

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

15x39

15x40

15x45*

15x49

15x50

15x51*

15x57

15x(57+11n)

15x65

15x70

15x71

15x81

15x86

16x18*

16x28

16x29

16x32

16x35

16x36

16x39

16x41

16x44

16x45

16x46

16x47

16x48

16x49

16x50

16x51

16x52

16x55

16x68

16x85

17x30

17x32

17x34

17x36

17x37

17x38

17x39

17x40

17x41

17x44

17x46

17x49

17x50

17x71

17x77

17x95

17x105

17x119

18x23

18x24

18x25

18x27

18x29

18x30*

18x30

18x31

18x31*

18x31

18x32

18x32*

18x32

18x34

18x35

18x35*

18x35

18x36

18x37

18x38

18x39*

18x39

18x40

18x41

18x42

18x43

18x44

18x45

18x46*

18x46

18x47

18x48

18x49

18x50

18x52

18x54

18x56

18x63*

18x70

18x71*

18x72 *

18x75

19x21

19x25

19x26

19x27

19x27

19x27

19x28

19x29

19x30

19x31

19x32

19x33

19x34

19x35

19x36

19x37

19x38

19x39

19x40

19x41

19x41

19x41

19x41

19x42

19x43

19x44

19x45

19x46

19x47

19x48

19x49

19x50

19x51

19x52

19x57

19x59

19x60

19x61

19x65

19x73

19x75*

19x77

19x104

20x24

20x24

20x26

20x27*

20x27

20x28

20x29*

20x30

20x31

20x32

20x33*

20x34

20x35

20x35

20x36

20x36*

20x36

20x37*

20x37

20x38

20x39

20x40*

20x40*

20x40

20x41

20x42

20x43

20x44

20x45*

20x45

20x46

20x47

20x48

20x49*

20x49

20x50

20x53

20x54

20x56

20x(56+22m)*

20x61

20x63 *

20x63

21x23

21x26

21x29

21x30*

21x30

21x31

21x32

21x33*

21x33

21x34

21x35

21x35*

21x36

21x37*

21x37

21x38*

21x38

21x39*

21x39

21x40

21x41*

21x41

21x42

21x43*

21x43

21x44

21x45

21x45*

21x45

21x45*

21x46

21x46*

21x46

21x47*

21x47

21x48

21x49

21x49*

21x49

21x50

21x51

21x55*

21x56

21x57*

21x66*

21x73

22x23

22x24

22x26

22x27*

22x28

22x28*

22x28

22x29

22x30*

22x31

22x32

22x33*

22x33

22x34

22x35*

22x36

22x37*

22x37

22x37*

22x38

22x39

22x40

22x40*

22x40

22x41

22x42

22x43

22x44

22x45

22x46*

22x46

22x47

22x48

22x49

22x50

22x53

22x56*

22x59

23x28

23x29

23x30

23x31*

23x32

23x33*

23x34

23x35

23x36

23x37

23x38

23x39

23x39*

23x39

23x40

23x41*

23x41

23x41*

23x41

23x42

23x43

23x44

23x45*

23x45

23x45

23x46

23x46

23x47

23x48

23x49

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

25x32

25x32*

25x32

25x33*

25x34*

25x34

25x36

25x38

25x39*

25x39

25x40*

25x41*

25x41

25x41*

25x41

25x42

25x43

25x44

25x45

25x45

25x45

25x45

25x46

25x47

25x48

25x49

25x50

25x57

26x28

26x29

26x30

26x31

26x33*

26x34

26x35

26x36*

26x36

26x37

26x38*

26x38

26x40

26x41

26x42*

26x42

26x43

26x44

26x45

26x46*

26x46

26x47