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

Lymnmeatlypamgfm 2022-05-03 Answered
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.
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Answers (1)

Aliana Sexton
Answered 2022-05-04 Author has 20 answers
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 R
which matches the accepted answer.
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