Out of curiosity I've been thinking about the following "puzzle" for a while now and maybe someone here can help.
We take a rectangle and start off at one of the corners. In that corner, which is 90 degrees, we start drawing a line at 45 degrees, splitting the corner into two equal parts and staying inside the rectangle with our line. As soon as the line hits an edge of the rectangle, we take a 90 degree "turn" so that we stay inside the rectangle and repeat this as often as we can.
My hypothesis is that we then eventually always end up in a(nother) corner, where our problem stops as we can't take a 90 degree turn there and stay inside the rectangle.
I've tried this in my head with several sizes of rectangles and it always works out, but I can't prove that it's always true for all rectangles. (also with non-integer sized rectangles, for example)
If there's anybody out there wanting to spend some time thinking about this, I would really be interested to find out the solution. :)
If we take squares, the proof is easy. Take a square with edges size 5 and give the bottom left corner the co-ordinate
(0,0). We start a line and end up immediately at
If we take a rectangle size 6 (x-axis) by 5 (y-axis), our line "bounces" at the following points:
(6,0) is of course a corner point.