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Out of curiosity I've been thinking about the following "puzzle" for a while now and maybe someone here can help.

Situation

We take a rectangle and start off at one of the corners. In that corner, which is $90^\circ$, we start drawing a line at $45^\circ$, 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 $90^\circ$ "turn" so that we stay inside the rectangle and repeat this as often as we can.

Question

My hypothesis is that we then eventually always end up in a (nother) corner, where our problem stops as we can't take a $90^\circ$ 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. :)

Example cases

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 (5, 5).

If we take a rectangle size 6 (x-axis) by 5 (y-axis), our line "bounces" at the following points: (0, 0);(5, 5); (6, 4); (2, 0); (0, 2); (3, 5); (6, 2); (4, 0); (0, 4); (1, 5); (6, 0) where (6, 0) is of course a corner point.

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    Nope, as I said, I'm looking for a proof that it's true for all rectangles, also with non-integer sizes. Or for a counter-example of course... But to analyze this in my head or with some paper, I use integer sides and for the few cases I did for my self, it always seems to work.2012-12-29

1 Answers 1

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Hint: Consider a tiling of the plane by these rectangles. When you want to "turn by $90^\circ$", think about the relationship of the turned line, with the continuation of the line. Use this to get a classification of the conditions when your line will intersect another corner.

Note: You reached your conclusion only because you considered very special rectangles. Find a rectangle where you never return to a corner.

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    Thanks :) Your "tiling" suggestion really set me off to the right path, I didn't think of that myself, so I accepted your answer.2012-12-29