by Serhiy Grabarchuk Jr
In each of the 6 horizontal rows, indicated with the left arrows, the ratio between the light and the dark moons’ colors equals 1:1. If imagine each full moon is divided into 8 equal segments, then in the 1st, 3rd and 5th columns the light:dark ratio is 12:0, while in the 2nd, 4th and 6th columns the light:dark ratio is 0:12.
How many moons it is necessary to rotate around their centers in order that the ratio between the light and the dark colors in each horizontal row and in each vertical column is always 1:1?
6 moons have to be rotated.
Proof. The number of light and dark segments in each column must be 6 and 6 respectively. If two moons are left intact in every “moon” column then the number of light segments in the even columns and the number of dark segments in the odd columns is 8 and 8 respectively – above the required number of 6. Thus, at most only one moon can remain intact in each column, two moons have to be rotated, or 6 in total.
One of the solutions is shown. The unmoved moons are indicated with the central dots.