Scanning Codes
A scanning code consists of a 7 by 7 grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of 90 degrees counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
The answer is 1022.
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1 solution
Draw a 7 by 7 square and analyze it. Since there are 4 given lines of symmetry (2 diagonals, 1 vertical, 1 horizontal) and they split the plot into 8 equivalent sections, we can take just one-eighth and study it in particular. Each of these sections has 10 distinct sub-squares, whether partially or in full. So since each can be colored either white or black, we choose 2^10=1024 but then subtract the 2 cases where all are white or all are black. That leaves us with 1022.
There are only ten squares we get to actually choose, and two independent choices for each, for a total of 2^10 = 1024 codes. Two codes must be subtracted (due to the rule that there must be at least one square of each color) for an answer of 1022.