How can Alex have so many crayons?
Alex colors each tile of a grid either white or black. A coloring is rotationally symmetric if the grid can be rotated or to achieve the same pattern. Two colorings are rotationally distinct if neither can be rotated to match the other.
How many rotationally distinct ways are there for Alex to color the grid such that the colorings are not rotationally symmetric ?
grid
This problem is not original.
The answer is 16320.
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1 solution
Each of the 1 6 cells is one of two colours; so there are 2 1 6 = 6 5 5 3 6 total colourings.
Any colouring with order 4 rotational symmetry also has order 2 rotational symmetry. So we just need to count how many have order 2 rotational symmetry.
These colourings are all fully defined by the top 4 × 2 block of 8 cells; so there are 2 8 = 2 5 6 of these.
Therefore there are 6 5 5 3 6 − 2 5 6 = 6 5 2 8 0 colourings with no rotational symmetry. However, each rotationally distinct colouring has been counted four times; so the total number of rotationally distinct colourings with no rotational symmetry is 6 5 2 8 0 ÷ 4 = 1 6 3 2 0 .