A stained glass window consists of nine squares of glass in a 3x3 array. Of the nine squares, k are red, the rest blue. A set of windows is produced such that any possible window can be formed in just one way by rotating and/or turning over one of the windows in the set. Altogether there are more than 100 red squares in the set. Find k.first, there are 8 Isometries of a square.Identity, three rotations (90,-90,180) four reflections (vertical, horizontal, two diagonal axis). let G be the permutation group, then |G|=8, and I can find fix(g) for every g.can someone give me a hint of how to proceed from there.

Question

Aliana Sexton · Accepted Answer

I upvoted the first answer but I would like to show how to compute the cycle index   Z  (  G  ) of the group   G of symmetries of the square and apply the Polya Enumeration Theorem to this problem.We need to enumerate and factor into cycles the eight permutations that contribute to   Z  (  G  )  .There is the identity, which contributes      a    1    9    .The two 90 degree rotations contribute  2      a    1        a    4    2    .The 180 degree rotation contributes      a    1        a    2    4    .The vertical and horizontal reflections contribute  2      a    1    3        a    2    3    .The reflections in a diagonal contribute  2      a    1    3        a    2    3    .This yields the cycle index  Z  (  G  )  =      1    8    (      a    1    9    +  2      a    1        a    4    2    +      a    1        a    2    4    +  4      a    1    3        a    2    3    )  .As we are interested in the red squares we evaluate  Z  (  G  )  (  1  +  R  )to get  1      /    8              (      1      +      R      )              9        +  1      /    2              (      1      +      R      )              3                  (                        R                          2                    +      1      )              3        +  1      /    8        (    1    +    R    )              (                        R                          2                    +      1      )              4          +  1      /    4        (    1    +    R    )              (                        R                          4                    +      1      )              2      which is            R              9        +  3              R              8        +  8              R              7        +  16              R              6        +  23              R              5        +  23              R              4        +  16              R              3        +  8              R              2        +  3    R  +  1.This is the classification of the orbits according to the number of red squares. Differentiate and multiply by   R to obtain the total count of the squares, which yields  9              R              9        +  24              R              8        +  56              R              7        +  96              R              6        +  115              R              5        +  92              R              4        +  48              R              3        +  16              R              2        +  3    Rwhich matches the accepted answer.

A stained glass window consists of nine squares of glass in a 3x3 array. Of the nine squares, k are

Answered question

Answer & Explanation

New Questions in High school probability