One 8x8 checkerboard is placed directly on top of another one so that their edges are aligned. The top checkerboard is then rotated 45 degrees about the common center of the two boards. What is the total area of the region where the black squares of the two boards overlap?
To make this easier to visualize, I made a diagram of the situation (click here)
. I started with yellow squares as the ones to be counted. In the rotation I made the color to be counted as transparent red. The colors might look different from conversion from png to jpg within paint, which has its own background and treated black as transparent.
But I noticed a symmetry: on the right side the top and bottom are complements of each other in a symmetric fashion. The overlaps of one match the misses of the other. So missed areas equal the hit areas.
Since the original chessboard has half the area to be counted, and half of that is lost as non-overlap, only 1/4 of the octagon where the boards overlap is the area we seek. This is easier to disentangle on two simplified 2x2 boards (click here for picture)
. There are eight kites that make up the octagonal overlap of the boards, only two of which are overlaps of the areas to be counted.
In the case of the two full 8x8 boards, the full octagon has a height of 8. As the octagon consists of 8 kites and each kite consists of two right triangles, the octagon consists of 16 right triangles. The longer side of each triangle is 4 and the acute angle in it is 22.5° so the shorter side is 4*tan(22.5°) ~= 1.65685424949238. The area of the triangle if 4 times this divided by 2, or 3.31370849898476. There are 16 of these triangles in the octagon, but we count only 1/4 of that total area, so we just multiply by 4: final answer, 13.254833995939.
Edited on August 22, 2019, 2:03 pm
Posted by Charlie
on 2019-08-22 11:24:22