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Color Tiles

Analysis and Copyright : ISHINO Keiichiro (2005). MacMahon's Color Tiles (Percy Alexander MacMahon) is famous color tiling puzzle. That is using 4 colored 24 equilateral triangles, 3 colored 24 squares and etc. Here, the problem of arranging 9 squares to 3×3 is taken up. This puzzle has many variations. Such as for instance: matches color on edges, matches color on vertexes, cannot rotate the pieces, can flip over the pieces, includes same color in a piece, exists identical piece, etc.

I think that the selection of the piece of this existing kind of puzzle is a random selection by the trial and error.

Sushi Puzzle
Kamon Puzzle

Matching 4 colors pair

When same color is not included in a piece, there are 96 kind of piece that uses 4 colors pair. Select different 9 pieces, and match color on edges. Surrounding colors are arbitrary. The pieces cannot be flipped over.

0 ABCD 1 ABDC 2 ACBD 3 ACDB 4 ADBC 5 ADCB
6 ABCd 7 ABDc 8 ACBd 9 ACDb10 ADBc11 ADCb
12 ABcD13 ABdC14 ACbD15 ACdB16 ADbC17 ADcB
18 ABcd19 ABdc20 ACbd21 ACdb22 ADbc23 ADcb
24 AbCD25 AbDC26 AcBD27 AcDB28 AdBC29 AdCB
30 AbCd31 AbDc32 AcBd33 AcDb34 AdBc35 AdCb
36 AbcD37 AbdC38 AcbD39 AcdB40 AdbC41 AdcB
42 Abcd43 Abdc44 Acbd45 Acdb46 Adbc47 Adcb
48 aBCD49 aBDC50 aCBD51 aCDB52 aDBC53 aDCB
54 aBCd55 aBDc56 aCBd57 aCDb58 aDBc59 aDCb
60 aBcD61 aBdC62 aCbD63 aCdB64 aDbC65 aDcB
66 aBcd67 aBdc68 aCbd69 aCdb70 aDbc71 aDcb
72 abCD73 abDC74 acBD75 acDB76 adBC77 adCB
78 abCd79 abDc80 acBd81 acDb82 adBc83 adCb
84 abcD85 abdC86 acbD87 acdB88 adbC89 adcB
90 abcd91 abdc92 acbd93 acdb94 adbc95 adcb

Color Tiles Piece A to D, a to d means a color, A and a, B and b, C and c, D and d are the same color pairs respectively. The 0th ABCD corresponds to the left figure.

4 Colors Pair Tiles It is possible to make it to the combination of small numbers as much as possible by substituting a suitable color for an arbitrary combination. For instance, 20 21 22 37 38 54 58 74 76 makes to 0 1 3 30 35 58 65 90 95 by substituting AaBbCcDd to CcaADdbB.

20 ACbd → 0 ABCD
21 ACdb → 3 ACDB
22 ADbc → 35 AdCb
37 AbdC → 1 ABDC
38 AcbD → 30 AbCd
54 aBCd → 58 aDBc
58 aDBc → 95 adcb
74 acBD → 90 abcd
76 adBC → 65 aDcB

Moreover, the color pair such as A and a is 12 necessary for the solution. When 9 chosen pieces don't have the pair 12, it doesn't have a solution obviously. The combination above has the pair 16.

The normalized combination that chooses 9 different from 96 kinds and has over 12 color pairs is as follows:

Total combinations 1,296,543,270,88096C9
Normalized combinations 2,911,418,829over 12 color pairs
Combinations with solution 648,813,123
Combinations with max solution 1655 solutions
Combinations with unique solution280,744,605

It is understood that the combination with unique solution is not too unusual. The combination with the maximum solution is as follows that has 655 solutions. In parentheses, the number of pairs is shown.

  1. 0 6 12 18 24 66 72 84 90 (16)

Yoshiaki Hirano instituted the following problems (2005/04):

I thought that such combinations did not exist. But, such 210 kinds of combinations exist about the followings: (Sushi Puzzle and Kamon Puzzle are one of such combinations)

  1. 0 1 2 20 23 45 54 93 94 (15)
  2. 0 1 2 20 45 47 54 70 93 (15)
  3. 0 1 2 20 45 47 70 78 87 (15)
  4. 0 1 2 22 56 72 81 93 95 (15)
  5. 0 1 2 37 46 47 70 78 87 (15)
  6. 0 1 2 37 46 47 72 87 94 (15)
  7. 0 1 3 31 64 67 85 87 95 (16)
  8. 0 1 3 32 70 71 78 82 85 (16)
  9. 0 1 3 33 34 69 85 87 95 (16)
  10. 0 1 3 34 44 68 79 83 84 (16)
  11. 0 1 6 11 65 72 80 89 93 (15)
  12. 0 1 6 20 45 70 82 87 95 (15)
  13. 0 1 6 21 47 65 78 79 87 (16)
  14. 0 1 6 23 39 62 70 82 93 (16)
  15. 0 1 6 39 47 58 68 93 94 (15)
  16. 0 1 6 39 47 62 70 82 93 (16)
  17. 0 1 8 11 22 74 85 87 89 (15)
  18. 0 1 8 14 23 45 54 82 93 (15)
  19. 0 1 8 14 23 45 78 82 87 (15)
  20. 0 1 8 14 45 47 54 58 93 (15)
  21. 0 1 8 14 45 47 58 78 87 (15)
  22. 0 1 8 14 71 73 87 94 95 (16)
  23. 0 1 8 16 36 70 87 89 93 (16)
  24. 0 1 8 17 37 46 70 78 93 (15)
  25. 0 1 8 18 21 71 81 82 91 (16)
  26. 0 1 8 19 46 71 77 81 90 (16)
  27. 0 1 8 22 25 80 83 89 93 (16)
  28. 0 1 8 22 33 72 87 89 94 (16)
  29. 0 1 8 22 33 84 87 88 89 (16)
  30. 0 1 8 22 35 61 74 93 95 (16)
  31. 0 1 8 22 35 74 85 87 89 (15)
  32. 0 1 8 22 37 50 87 89 95 (15)
  33. 0 1 8 22 47 50 85 87 89 (15)
  34. 0 1 8 23 30 79 87 89 90 (16)
  35. 0 1 8 23 37 46 70 78 87 (15)
  36. 0 1 8 23 37 46 72 87 94 (15)
  37. 0 1 8 23 38 45 54 69 82 (15)
  38. 0 1 8 23 38 45 63 78 82 (15)
  39. 0 1 8 23 39 62 70 73 95 (15)
  40. 0 1 8 23 44 45 48 69 82 (15)
  41. 0 1 8 23 44 45 63 72 82 (15)
  42. 0 1 8 23 45 49 70 82 95 (16)
  43. 0 1 8 24 46 70 75 89 93 (16)
  44. 0 1 8 25 46 75 80 83 89 (15)
  45. 0 1 8 27 54 70 82 85 95 (15)
  46. 0 1 8 27 54 73 82 94 95 (15)
  47. 0 1 8 30 38 58 63 93 95 (16)
  48. 0 1 8 35 46 73 75 80 89 (15)
  49. 0 1 8 35 46 74 75 85 89 (15)
  50. 0 1 8 37 46 50 75 89 95 (15)
  51. 0 1 8 37 46 74 75 83 89 (15)
  52. 0 1 8 38 45 47 51 58 78 (15)
  53. 0 1 8 38 45 47 58 63 78 (15)
  54. 0 1 8 41 62 70 72 75 93 (15)
  55. 0 1 8 44 45 48 58 69 95 (16)
  56. 0 1 8 45 47 52 55 70 95 (16)
  57. 0 1 8 45 47 58 65 73 94 (16)
  58. 0 1 8 45 47 58 65 79 88 (16)
  59. 0 1 8 45 47 61 75 84 91 (16)
  60. 0 1 8 46 47 50 75 85 89 (15)
  61. 0 1 8 47 49 58 71 93 94 (16)
  62. 0 1 8 47 54 70 75 82 85 (15)
  63. 0 1 8 47 54 73 75 82 94 (15)
  64. 0 1 9 14 46 74 77 89 90 (16)
  65. 0 1 9 28 41 84 87 88 92 (16)
  66. 0 1 9 28 43 54 70 77 86 (15)
  67. 0 1 9 28 43 58 77 78 86 (15)
  68. 0 1 9 31 44 63 75 91 95 (16)
  69. 0 1 9 31 45 76 85 87 89 (16)
  70. 0 1 9 39 75 82 85 93 95 (16)
  1. 0 1 9 41 55 72 80 92 94 (16)
  2. 0 1 9 44 45 54 58 65 83 (16)
  3. 0 1 12 35 45 75 84 85 89 (16)
  4. 0 1 12 35 46 47 50 71 90 (15)
  5. 0 1 12 39 44 64 68 77 93 (16)
  6. 0 1 12 39 44 64 68 89 93 (16)
  7. 0 1 12 44 45 58 68 87 95 (15)
  8. 0 1 14 22 56 72 75 93 95 (15)
  9. 0 1 14 22 68 72 75 87 95 (15)
  10. 0 1 14 23 34 45 56 75 78 (15)
  11. 0 1 14 23 39 63 66 70 91 (16)
  12. 0 1 14 23 54 70 82 85 93 (16)
  13. 0 1 14 27 37 67 73 93 95 (16)
  14. 0 1 14 27 42 73 82 87 95 (16)
  15. 0 1 14 33 54 70 82 85 89 (16)
  16. 0 1 14 34 37 41 72 82 93 (15)
  17. 0 1 14 34 37 72 75 82 95 (16)
  18. 0 1 14 47 68 70 72 75 87 (15)
  19. 0 1 15 18 47 62 70 87 94 (16)
  20. 0 1 15 20 46 66 70 87 95 (15)
  21. 0 1 15 22 32 36 73 80 95 (15)
  22. 0 1 15 22 60 64 68 89 93 (16)
  23. 0 1 15 28 44 65 70 78 92 (16)
  24. 0 1 15 32 36 46 56 71 79 (15)
  25. 0 1 15 32 36 46 71 73 80 (15)
  26. 0 1 15 32 36 70 73 80 95 (16)
  27. 0 1 15 32 36 71 73 80 94 (16)
  28. 0 1 15 32 37 62 70 87 94 (16)
  29. 0 1 15 41 46 48 79 83 93 (16)
  30. 0 1 15 44 45 52 66 73 95 (16)
  31. 0 1 15 46 47 66 68 70 87 (15)
  32. 0 1 18 27 45 64 68 89 92 (16)
  33. 0 1 20 21 23 32 48 82 93 (15)
  34. 0 1 20 21 23 32 72 82 87 (15)
  35. 0 1 20 21 32 48 58 93 95 (16)
  36. 0 1 20 21 32 48 71 82 93 (16)
  37. 0 1 20 21 32 55 64 84 95 (16)
  38. 0 1 20 21 32 58 60 93 95 (16)
  39. 0 1 20 23 26 51 67 84 88 (15)
  40. 0 1 20 23 32 45 63 72 94 (15)
  41. 0 1 20 23 34 50 75 85 89 (15)
  42. 0 1 20 23 50 58 73 93 95 (15)
  43. 0 1 20 24 34 58 75 89 93 (15)
  44. 0 1 20 24 34 58 75 93 95 (16)
  45. 0 1 20 30 34 58 75 93 95 (16)
  46. 0 1 20 32 45 48 69 70 95 (16)
  47. 0 1 20 33 49 67 79 87 95 (16)
  48. 0 1 20 34 37 62 75 89 95 (16)
  49. 0 1 20 35 36 51 71 82 87 (16)
  50. 0 1 20 36 39 58 69 80 95 (16)
  51. 0 1 20 39 46 54 58 81 95 (16)
  52. 0 1 21 26 31 63 81 91 95 (16)
  53. 0 1 21 28 37 53 66 70 86 (15)
  54. 0 1 21 31 47 66 75 83 84 (16)
  55. 0 1 21 32 41 55 64 68 95 (16)
  56. 0 1 21 32 44 52 55 72 95 (16)
  57. 0 1 21 32 44 55 64 72 95 (16)
  58. 0 1 21 34 36 44 65 73 92 (15)
  59. 0 1 21 36 44 65 73 82 92 (16)
  60. 0 1 21 41 45 58 64 68 72 (15)
  61. 0 1 21 42 61 71 75 78 89 (16)
  62. 0 1 21 45 52 56 70 72 95 (15)
  63. 0 1 21 45 52 68 70 72 89 (15)
  64. 0 1 24 27 41 71 80 85 93 (16)
  65. 0 1 24 34 44 58 75 77 93 (15)
  66. 0 1 24 34 44 58 75 83 93 (16)
  67. 0 1 24 34 58 68 81 87 95 (16)
  68. 0 1 24 42 47 56 65 84 93 (16)
  69. 0 1 26 33 69 73 87 89 91 (16)
  70. 0 1 26 34 47 59 69 79 92 (16)
  1. 0 1 26 34 57 83 84 87 92 (16)
  2. 0 1 26 41 45 51 78 86 94 (16)
  3. 0 1 27 30 47 56 62 70 93 (16)
  4. 0 1 27 34 45 55 64 68 92 (16)
  5. 0 1 27 34 46 54 68 73 95 (16)
  6. 0 1 27 34 46 54 73 80 83 (15)
  7. 0 1 27 34 46 54 73 83 92 (16)
  8. 0 1 27 34 54 68 70 81 95 (16)
  9. 0 1 30 41 46 64 68 81 87 (16)
  10. 0 1 32 35 44 55 69 70 72 (16)
  11. 0 1 32 35 54 73 75 82 94 (15)
  12. 0 1 32 35 56 73 75 89 94 (15)
  13. 0 1 32 36 63 70 73 80 83 (16)
  14. 0 1 33 39 42 52 68 77 91 (16)
  15. 0 1 33 46 55 72 77 80 92 (16)
  16. 0 1 36 39 44 53 73 92 94 (16)
  17. 0 1 36 44 52 68 77 79 93 (16)
  18. 0 1 39 47 61 64 68 71 82 (16)
  19. 0 5 7 22 30 67 68 87 88 (16)
  20. 0 5 8 19 65 68 79 81 88 (16)
  21. 0 5 18 21 43 64 74 79 82 (16)
  22. 0 5 18 21 44 65 72 82 88 (16)
  23. 0 5 18 21 56 65 70 83 91 (16)
  24. 0 5 18 22 35 68 71 75 91 (16)
  25. 0 5 18 35 44 72 75 82 86 (16)
  26. 0 5 18 40 44 62 67 75 85 (16)
  27. 0 5 18 40 62 69 84 89 92 (16)
  28. 0 5 18 47 64 74 78 81 94 (16)
  29. 0 5 20 43 74 81 85 87 94 (16)
  30. 0 6 7 24 32 44 51 64 88 (14)
  31. 0 6 9 17 43 61 74 83 84 (16)
  32. 0 6 10 20 23 64 77 87 90 (16)
  33. 0 6 10 38 41 56 64 71 78 (16)
  34. 0 6 11 27 43 50 70 82 93 (16)
  35. 0 6 11 27 50 67 79 88 92 (16)
  36. 0 6 11 31 39 70 85 89 93 (16)
  37. 0 6 11 39 55 68 70 80 91 (16)
  38. 0 6 11 45 56 58 79 87 94 (16)
  39. 0 6 11 46 56 64 79 89 92 (16)
  40. 0 6 16 27 55 59 68 80 91 (16)
  41. 0 6 16 27 56 67 79 83 92 (16)
  42. 0 6 17 33 42 53 67 88 95 (16)
  43. 0 6 17 43 53 62 78 82 95 (16)
  44. 0 6 17 46 56 64 74 83 92 (16)
  45. 0 6 22 39 55 59 68 80 91 (16)
  46. 0 6 23 25 45 50 58 87 94 (16)
  47. 0 6 23 27 37 56 58 81 94 (16)
  48. 0 6 23 27 43 73 77 81 87 (16)
  49. 0 6 23 45 56 58 73 87 94 (16)
  50. 0 6 24 26 31 43 52 63 69 (14)
  51. 0 6 27 36 43 53 61 69 95 (16)
  52. 0 7 8 23 37 46 72 87 88 (15)
  53. 0 7 8 35 43 54 64 87 92 (15)
  54. 0 7 8 36 45 52 68 79 89 (16)
  55. 0 7 8 36 47 59 69 74 76 (16)
  56. 0 7 9 13 18 44 51 79 82 (14)
  57. 0 7 9 13 18 44 51 82 85 (14)
  58. 0 7 9 13 23 45 51 88 91 (15)
  59. 0 7 9 13 23 45 63 79 80 (15)
  60. 0 7 9 34 54 59 80 81 95 (16)
  61. 0 7 9 35 56 65 66 78 95 (16)
  62. 0 7 9 41 43 54 64 68 95 (16)
  63. 0 7 11 18 51 67 81 85 95 (16)
  64. 0 7 11 37 39 54 64 89 93 (16)
  65. 0 7 11 39 42 56 73 81 95 (16)
  66. 0 7 13 23 30 59 69 75 90 (16)
  67. 0 7 14 32 46 49 63 81 95 (16)
  68. 0 7 15 45 52 68 70 72 89 (16)
  69. 0 7 21 29 43 66 70 75 83 (16)
  70. 0 7 29 44 59 61 69 75 84 (16)

The highlight combinations are better than others.

When the pieces can be flipped over, there are 48 kind of piece.

0 ABCD 1 ABDC 2 ACBD
6 ABCd 7 ABDc 8 ACBd 9 ACDb10 ADBc11 ADCb
12 ABcD13 ABdC14 ACbD
18 ABcd19 ABdc20 ACbd21 ACdb22 ADbc23 ADcb
30 AbCd31 AbDc32 AcBd
42 Abcd43 Abdc44 Acbd
48 aBCD49 aBDC50 aCBD
54 aBCd55 aBDc56 aCBd57 aCDb58 aDBc59 aDCb
60 aBcD61 aBdC62 aCbD
66 aBcd67 aBdc68 aCbd69 aCdb70 aDbc71 aDcb
78 abCd79 abDc80 acBd
90 abcd91 abdc92 acbd

The normalized combination that chooses 9 different from 48 kinds is as follows:

Total combinations 1,677,106,64048C9
Normalized combinations 3,956,251over 12 color pairs
Combinations with solution 3,572,154
Combinations with max solution 19,812 solutions
Combinations with unique solution70,803

The combination with the maximum solution is as follows that has 9,812 solutions.

  1. 0 6 11 18 23 60 66 71 78 (16)

The combination that have unique solution even if all 9 pieces place the center is the following 1 combination. ☞ 2-side Match

  1. 0 1 6 18 21 44 50 66 91 (13)

 

Matching 5 colors

When same color is not included in a piece, there are 30 kind of piece that uses 5 colors. Select different 9 pieces, and match color on edges. Surrounding colors are arbitrary. The pieces cannot be flipped over.

0 ABCD 1 ABDC 2 ACBD 3 ACDB 4 ADBC 5 ADCB
6 ABCE 7 ABEC 8 ACBE 9 ACEB10 AEBC11 AECB
12 ABDE13 ABED14 ADBE15 ADEB16 AEBD17 AEDB
18 ACDE19 ACED20 ADCE21 ADEC22 AECD23 AEDC
24 BCDE25 BCED26 BDCE27 BDEC28 BECD29 BEDC

Color Tiles Piece A to E means a color. The 0th ABCD corresponds to the left figure.

5 Colors Tiles It is possible to make it to the combination of small numbers as much as possible by substituting a suitable color for an arbitrary combination. For instance, 2 6 8 15 16 18 20 24 26 makes to 0 1 6 7 14 15 20 21 24 by substituting ABCDE to BDCEA.

2 ACBD → 24 BCDE
6 ABCE → 1 ABDC
8 ACBE → 0 ABCD
15 ADEB → 14 ADBE
16 AEBD → 15 ADEB
18 ACDE → 6 ABCE
20 ADCE → 7 ABEC
24 BCDE → 20 ADCE
26 BDCE → 21 ADEC

The normalized combination that chooses 9 different from 30 kinds is as follows:

Total combinations 14,307,15030C9
Normalized combinations 119,507
Combinations with solution 119,505
Combinations with max solution 2763 solutions
Combinations with min solution 54 solutions

The combinations with the maximum solution are as follows that has 763 solutions.

  1. 0 1 2 3 4 5 6 7 9
  2. 0 1 2 3 4 5 6 7 11

The combinations with the minimum solution are as follows that has 4 solutions. There are no unique solutions.

  1. 0 1 2 3 7 12 13 16 26
  2. 0 1 2 6 9 16 18 19 26
  3. 0 1 2 6 12 13 16 19 27
  4. 0 1 2 7 10 12 21 22 27
  5. 0 1 3 6 10 16 19 24 25

The combinations without a solution are the followings:

  1. 0 1 2 3 12 13 16 19 22
  2. 0 1 2 6 7 10 12 13 14

The combinations that have unique solution even if all 9 pieces place the center are not exist.

When the pieces can be flipped over, there are 15 kind of piece.

0 ABCD 1 ABDC 2 ACBD
6 ABCE 7 ABEC 8 ACBE
12 ABDE13 ABED14 ADBE
18 ACDE19 ACED20 ADCE
24 BCDE25 BCED26 BDCE

The normalized combination that chooses 9 different from 15 kinds is as follows:

Total combinations 5,00515C9
Normalized combinations 58
Combinations with solution 58
Combinations with max solution1 30,228 solutions
Combinations with min solution1 17,872 solutions

The combinations that have unique solution even if all 9 pieces place the center are not exist.

Matching 4 colors

When same color may be included in a piece, there are 70 kind of piece that uses 4 colors. Select different 9 pieces, and match color on edges. Surrounding colors are arbitrary. The pieces cannot be flipped over.

0 ABCD 1 ABDC 2 ACBD 3 ACDB 4 ADBC 5 ADCB
6 AABC 7 AACB 8 AABD 9 AADB10 AACD11 AADC
12 ACBB13 ABBC14 ADBB15 ABBD16 BBCD17 BBDC
18 ABCC19 ACCB20 ADCC21 ACCD22 BDCC23 BCCD
24 ABDD25 ADDB26 ACDD27 ADDC28 BCDD29 BDDC
30 ABAC31 ABAD32 ACAD33 ABCB34 ABDB35 BCBD
36 ACBC37 ACDC38 BCDC39 ADBD40 ADCD41 BDCD
42 AABB43 AACC44 AADD45 BBCC46 BBDD47 CCDD
48 ABAB49 ACAC50 ADAD51 BCBC52 BDBD53 CDCD
54 AAAB55 AAAC56 AAAD57 ABBB58 BBBC59 BBBD
60 ACCC61 BCCC62 CCCD63 ADDD64 BDDD65 CDDD
66 AAAA67 BBBB68 CCCC69 DDDD

Color Tiles Piece A to D means a color. The 0th ABCD corresponds to the left figure.

4 Colors Tiles It is possible to make it to the combination of small numbers as much as possible by substituting a suitable color for an arbitrary combination. For instance, 1 3 23 36 38 39 41 42 69 makes to 0 5 6 30 32 33 35 47 67 by substituting ABCD to DCAB.

1 ABDC → 5 ADCB
3 ACDB → 0 ABCD
23 BCCD → 6 AABC
36 ACBC → 32 ACAD
38 BCDC → 30 ABAC
39 ADBD → 35 BCBD
41 BDCD → 33 ABCB
42 AABB → 47 CCDD
69 DDDD → 67 BBBB

The normalized combination that chooses 9 different from 70 kinds is as follows:

Total combinations 65,033,528,56070C9
Normalized combinations 2,713,252,941
Combinations with solution2,590,002,702
Combinations with unique solution31,611,564

The combinations that have unique solution even if all 9 pieces place the center are not exist.

When the pieces can be flipped over, there are 55 kind of piece.

0 ABCD 1 ABDC 2 ACBD
6 AABC 8 AABD10 AACD
12 ACBB14 ADBB16 BBCD
18 ABCC20 ADCC22 BDCC
24 ABDD26 ACDD28 BCDD
30 ABAC31 ABAD32 ACAD33 ABCB34 ABDB35 BCBD
36 ACBC37 ACDC38 BCDC39 ADBD40 ADCD41 BDCD
42 AABB43 AACC44 AADD45 BBCC46 BBDD47 CCDD
48 ABAB49 ACAC50 ADAD51 BCBC52 BDBD53 CDCD
54 AAAB55 AAAC56 AAAD57 ABBB58 BBBC59 BBBD
60 ACCC61 BCCC62 CCCD63 ADDD64 BDDD65 CDDD
66 AAAA67 BBBB68 CCCC69 DDDD

The normalized combination that chooses 9 different from 55 kinds is as follows:

Total combinations 6,358,402,05055C9
Normalized combinations 266,787,966
Combinations with solution 244,911,124
Combinations with unique solution1,001,305

The combinations that have unique solution even if all 9 pieces place the center are not exist.

Matching 3 colors

When same color may be included in a piece, there are 24 kind of piece that uses 3 colors. This is a subset of 4 colors set. Select different 9 pieces, and match color on edges. Surrounding colors are arbitrary. The pieces cannot be flipped over.

0 AABC 1 AACB 2 ACBB 3 ABBC 4 ABCC 5 ACCB
6 ABAC 7 ABCB 8 ACBC 9 AABB10 AACC11 BBCC
12 ABAB13 ACAC14 BCBC15 AAAB16 AAAC17 ABBB
18 BBBC19 ACCC20 BCCC21 AAAA22 BBBB23 CCCC

Color Tiles Piece A to C means a color. The 0th AABC corresponds to the left figure.

3 Colors Tiles It is possible to make it to the combination of small numbers as much as possible by substituting a suitable color for an arbitrary combination. For instance, 1 2 3 13 16 17 18 21 22 makes to 0 1 3 14 15 16 18 21 22 by substituting ABC to BAC.

1 AACB → 3 ABBC
2 ACBB → 0 AABC
3 ABBC → 1 AACB
13 ACAC → 14 BCBC
16 AAAC → 18 BBBC
17 ABBB → 15 AAAB
18 BBBC → 16 AAAC
21 AAAA → 22 BBBB
22 BBBB → 21 AAAA

The normalized combination that chooses 9 different from 24 kinds is as follows:

Total combinations 1,307,50424C9
Normalized combinations 218,596
Combinations with solution215,734
Combinations with max solution 125,488 solutions
Combinations with unique solution56

The combination with the maximum solution is as follows that has 25,488 solutions.

  1. 0 1 2 3 4 5 6 7 8

The combinations with the unique solution are as follows.

  1. 0 1 2 4 14 15 20 21 23
  2. 0 1 2 8 13 14 16 21 22
  3. 0 1 2 8 13 14 18 21 22
  4. 0 1 2 8 13 16 18 21 22
  5. 0 1 2 8 14 16 18 21 22
  6. 0 1 2 13 14 15 18 21 22
  7. 0 1 2 13 16 17 18 21 22
  8. 0 1 2 14 15 16 18 21 22
  9. 0 1 3 5 14 15 20 21 23
  10. 0 1 3 8 13 14 16 21 22
  11. 0 1 3 8 13 14 18 21 22
  12. 0 1 3 8 13 16 18 21 22
  13. 0 1 3 8 14 16 18 21 22
  14. 0 1 3 13 14 15 18 21 22
  15. 0 1 3 13 16 17 18 21 22
  16. 0 1 3 14 15 16 18 21 22
  17. 0 2 4 7 12 14 15 21 23
  18. 0 2 4 7 12 14 20 21 23
  19. 0 2 4 7 12 15 20 21 23
  1. 0 2 4 7 14 15 20 21 23
  2. 0 2 4 12 17 19 20 22 23
  3. 0 2 4 14 15 16 18 21 22
  4. 0 2 6 8 14 16 18 21 22
  5. 0 3 10 11 12 15 21 22 23
  6. 0 3 10 11 12 17 21 22 23
  7. 0 3 10 12 14 15 21 22 23
  8. 0 3 10 12 14 17 21 22 23
  9. 0 3 10 12 15 18 21 22 23
  10. 0 3 10 12 15 20 21 22 23
  11. 0 3 10 12 17 18 21 22 23
  12. 0 3 10 12 17 20 21 22 23
  13. 0 3 11 12 13 15 21 22 23
  14. 0 3 11 12 13 17 21 22 23
  15. 0 3 11 12 15 16 21 22 23
  16. 0 3 11 12 15 19 21 22 23
  17. 0 3 11 12 16 17 21 22 23
  18. 0 3 11 12 17 19 21 22 23
  19. 0 3 12 13 14 15 21 22 23
  1. 0 3 12 13 14 17 21 22 23
  2. 0 3 12 13 15 18 21 22 23
  3. 0 3 12 13 15 20 21 22 23
  4. 0 3 12 13 17 18 21 22 23
  5. 0 3 12 13 17 20 21 22 23
  6. 0 3 12 14 15 16 21 22 23
  7. 0 3 12 14 15 19 21 22 23
  8. 0 3 12 14 16 17 21 22 23
  9. 0 3 12 14 17 19 21 22 23
  10. 0 3 12 15 16 18 21 22 23
  11. 0 3 12 15 16 20 21 22 23
  12. 0 3 12 15 18 19 21 22 23
  13. 0 3 12 15 19 20 21 22 23
  14. 0 3 12 16 17 18 21 22 23
  15. 0 3 12 16 17 20 21 22 23
  16. 0 3 12 17 18 19 21 22 23
  17. 0 3 12 17 19 20 21 22 23
  18. 0 5 6 7 14 15 20 21 23

The combinations that have unique solution even if all 9 pieces place the center are not exist.

MacMahon's Color Tiles is using all 24 pieces and constructs 6×4. Then surrounding colors are arranged.

When the pieces can be flipped over, there are 21 kind of piece.

0 AABC 2 ACBB 4 ABCC
6 ABAC 7 ABCB 8 ACBC 9 AABB10 AACC11 BBCC
12 ABAB13 ACAC14 BCBC15 AAAB16 AAAC17 ABBB
18 BBBC19 ACCC20 BCCC21 AAAA22 BBBB23 CCCC

The normalized combination that chooses 9 different from 21 kinds is as follows:

Total combinations 293,93021C9
Normalized combinations 49,469
Combinations with solution 47,690
Combinations with max solution 1 128,224 solutions
Combinations with min solution 801 4 solutions

The combinations that have unique solution even if all 9 pieces place the center are not exist.

May 12, 2005 by k16@chiba.email.ne.jp