A circular wheel has 7 equal-sized sections, and each of them will be painted with one of two colors. Two colorings are considered equivalent if one can be rotated to produce the other. In how many non-equivalent ways can the wheel be painted?

![](https://cdn.mathpix.com/cropped/2024_05_01_4fd1b602786ad9981479g-32.jpg?height=338&width=347&top_left_y=2028&top_left_x=736)

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Solution

Without considering rotations, since each sector admits two possibilities of painting, there are $2^{7}$ fillings of the 7 sections with two colors. Each of these fillings can be rotated in 6 ways to generate 6 other equivalent fillings. For example, if the colors are represented by $A$ and $B$, then the following paintings are equivalent:

![](https://cdn.mathpix.com/cropped/2024_05_01_4fd1b602786ad9981479g-33.jpg?height=362&width=364&top_left_y=718&top_left_x=693)

$A B A A A A B$

![](https://cdn.mathpix.com/cropped/2024_05_01_4fd1b602786ad9981479g-33.jpg?height=362&width=367&top_left_y=718&top_left_x=1154)

$A A A B A B A$

In the drawing above, each painting is associated with a word that represents the reading of the colors in a clockwise direction, and the complete list of equivalent paintings is:

$$
\begin{aligned}
& A B A A A A B \\
& B A A A A B A \\
& A A A A B A B \\
& A A A B A B A \\
& A A B A B A A \\
& A B A B A A A \\
& B A B A A A A
\end{aligned}
$$

These fillings obtained by rotation will be distinct if not all the chosen colors are the same. There are only two paintings where all sections have the same color. Removing these from the total, the rest can be grouped into sets of 7 equivalent paintings. Therefore, the total number of non-equivalent paintings is

$$
\frac{2^{7}-2}{7}+2=20
$$