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The Mystery of the Sealed Box
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Our questions fall into several categories.
- Corners. If a path starts at a corner, does it hit a corner?
Or does it go on forever? And if it hits a corner, which corner? Clearly, the answer depends on the dimensions of the box, but how? We have some answers.
- Loops?. If the box has integral dimensions,
are there loops?
Actually, every box has loops; the interesting question is: Are there loops that meet edges at integral distances from the corners? For some boxes, even big boxes, there are no such loops; every path leads to a corner.
- Number theory. The context of the rectangle suggests odd generalizations of "least common multiple", "greatest common divisor", and "relatively prime" for triples. We have worked a lot on one possibility. In our scheme, 3,4,5 are not relatively prime. But 3,4,6 are.
- Analysis. We have what we call a "generalized continued fraction". It yields pairs of numbers. It's connected to the fractal.
