Tiling Problem with Patterns and Colors












1














My wife proposed an interesting tiling problem to me. The specific problem she proposed is:



I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size 6 by 7, where "the patterns don't touch", meaning,




  • The same pattern can not touch at all

  • Like colors, but different patterns can touch only on a diagonal


I have trouble encoding those two rules into a relationship I can work with. Can someone help me solve this specific problem, and frame it in a general way?






combinatorics tiling







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  • 1




    There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
    – Ross Millikan
    Jan 4 at 3:55










  • One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
    – Anders Kaseorg
    Jan 4 at 5:24
















1














My wife proposed an interesting tiling problem to me. The specific problem she proposed is:



I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size 6 by 7, where "the patterns don't touch", meaning,




  • The same pattern can not touch at all

  • Like colors, but different patterns can touch only on a diagonal


I have trouble encoding those two rules into a relationship I can work with. Can someone help me solve this specific problem, and frame it in a general way?






combinatorics tiling







share|cite|improve this question


















  • 1




    There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
    – Ross Millikan
    Jan 4 at 3:55










  • One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
    – Anders Kaseorg
    Jan 4 at 5:24














1












1








1







My wife proposed an interesting tiling problem to me. The specific problem she proposed is:



I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size 6 by 7, where "the patterns don't touch", meaning,




  • The same pattern can not touch at all

  • Like colors, but different patterns can touch only on a diagonal


I have trouble encoding those two rules into a relationship I can work with. Can someone help me solve this specific problem, and frame it in a general way?






combinatorics tiling







share|cite|improve this question













My wife proposed an interesting tiling problem to me. The specific problem she proposed is:



I have tiles of 6 different patterns. Each pattern is in 3 different colors. I want to make a quilt of size 6 by 7, where "the patterns don't touch", meaning,




  • The same pattern can not touch at all

  • Like colors, but different patterns can touch only on a diagonal


I have trouble encoding those two rules into a relationship I can work with. Can someone help me solve this specific problem, and frame it in a general way?






combinatorics tiling




combinatorics tiling






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Jan 4 at 3:41









user79950user79950

154110




154110








  • 1




    There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
    – Ross Millikan
    Jan 4 at 3:55










  • One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
    – Anders Kaseorg
    Jan 4 at 5:24














  • 1




    There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
    – Ross Millikan
    Jan 4 at 3:55










  • One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
    – Anders Kaseorg
    Jan 4 at 5:24








1




1




There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
– Ross Millikan
Jan 4 at 3:55




There are $18$ combinations of pattern and color. Do you want to use $2$ or $3$ of each?
– Ross Millikan
Jan 4 at 3:55












One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
– Anders Kaseorg
Jan 4 at 5:24




One solution is to make a 2×2 square of {{A1, B2}, {C2, D1}} and tile it to any desired size, using only four patterns A, B, C, D and two colors 1, 2. This follows your rules but probably isn’t the solution you want; if so, can you make a rule that explains why it isn’t the solution you want?
– Anders Kaseorg
Jan 4 at 5:24










1 Answer
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You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.



A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3

C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1

E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2





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    You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.



    A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3
    
C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1
    
E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2





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      3














      You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.



      A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3
      
C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1
      
E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2





      share|cite|improve this answer
























        3












        3








        3






        You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.



        A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3
        
C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1
        
E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2





        share|cite|improve this answer












        You do not need to consider both rules at the same time. Tile the patterns (A,B,C,D,E,F) so that the first rule is satisfied. Tile the colors (1,2,3) so that the second rule is satisfied. Then combine the two to get the desired result.



        A B C D E F     1 2 3 1 2 3     A1 B2 C3 D1 E2 F3
        
C D E F A B  +  2 3 1 2 3 1  =  C2 D3 E1 F2 A3 B1
        
E F A B C D     3 1 2 3 1 2     E3 F1 A2 B3 C1 D2






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        answered Jan 4 at 12:28









        Daniel MathiasDaniel Mathias

        3306




        3306






























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