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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 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.

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 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. :)

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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Are you assuming that the side lengths of your rectangle are integers? –  Hans Lundmark Dec 29 '12 at 12:45
    
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. –  bartlaarhoven Dec 29 '12 at 12:47

1 Answer 1

up vote 4 down vote accepted

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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Ah, tiling them, that's a good one. But that means that, for this to work, the two sides should have a common divisor, right? And that would mean that it would work for any rectangle with sides from $\mathbb Q$ but not for example for rectangles size 1 by $\pi$ ? –  bartlaarhoven Dec 29 '12 at 12:54
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There are sufficient details for you to work this out for yourself. Be careful with the term 'common divisor', which mainly applies to integers only. For example, does $3\sqrt{2}$ and $4\sqrt{2}$ have a 'common divisor'? Think carefully about what the condition that you want actually is, and how you can see that from the picture. –  Calvin Lin Dec 29 '12 at 12:57
    
Hm. How do you call that then.. For me $3\sqrt{2}$ and $4\sqrt{2}$ have a "common divisor", being $\sqrt{2}$. And if I do this one on paper, it also works out, you end up in another corner. But the rectangle of 1 by $\pi$ will never work out, just like the rectangle of 3 by $\sqrt{2}$, right? Because you can't divide both sides into an integer number of equally sized parts... (whatever that may be called) –  bartlaarhoven Dec 29 '12 at 13:03
    
My point was that you have to clarify what 'common divisor' means. Generally divisors are integers, so saying that the 'common divisor' is $\sqrt{2}$ needs a bit of explanation. For example, would the $\frac {22}{7} \times \frac {7}{22}$ rectangle lead you back to a corner? More importantly, you should be able to classify all rectangles which will lead you back to a corner. –  Calvin Lin Dec 29 '12 at 13:16
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I guess a $a\times b$ rectangle satisfies the condition if and only if there exist integers $x,y$ such that $xa=by$. But i'm not sure about that. –  barto Dec 29 '12 at 13:20

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