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TIFR2021-Maths-A: 10

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Let $\mathcal{C}$ denote the set of colorings of an $8\times 8$ chessboard, where each square is colored either black or white. Let $\thicksim$ denote the equivalence relation on $\mathcal{C}$ defined as follows: two colorings are equivalent if and only if one of them can be obtained from the other by a rotation of the chessboard. The cardinality of the set $\mathcal{C}/\thicksim$ of equivalence classes of elements of $\mathcal{C}$ under $\thicksim$ is

  1. $2^{62}$
  2. $2^{62}+2^{30}+2^{15}$
  3. $2^{64}-2^{32}+2^{16}$
  4. $2^{63}-2^{31}+2^{15}$
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