Problem 6. Master Li Si Qing makes fans. Each fan consists of 6 sectors, painted on both sides in red and blue (see fig.). Moreover, if one side of a sector is painted red, the opposite side is painted blue and vice versa. Any two fans made by the master differ in coloring (if one coloring can be transformed into another by flipping the fan, they are considered the same). What is the maximum number of fans the master can make?
Answer: 36.

## Solution:

The coloring of one side can be chosen in $2^{6}=64$ ways. It uniquely determines the coloring of the opposite side. However, some fans - those that transform into each other when flipped, we have counted twice. To find their number, let's see how many fans transform into themselves when flipped. There are 8 such fans in total. So, we need to divide the remaining 64 $8=56$ by two - which gives us 28. We need to add those 8 that transform into themselves.