The Three Red Squares.
Jack, the painter, wants to paint three squares of a grid with red color. What is the number of ways in which he can do so such that each red square shares at least one vertex with at least one red square?
The answer is 1840.
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1 solution
There are only 5 possible configurations of squares, up to reflection and rotation:
Red: 4 rotations × 1 0 ⋅ 1 0 positions = 4 0 0
Blue: 4 rotations × 2 reflections × 9 ⋅ 1 0 positions = 7 2 0
Green: 2 rotations × 9 ⋅ 1 1 positions = 1 9 8
Yellow: 4 rotations × 9 ⋅ 1 0 positions = 3 6 0
Purple: 2 rotations × 9 ⋅ 9 positions = 1 6 2
4 0 0 + 7 2 0 + 1 9 8 + 3 6 0 + 1 6 2 = 1 8 4 0