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 Dao Juxing (The way of the rectangle) (Posted on 2008-03-28)

Show that the number of simply connected closed paths through a 2xM rectangle grows at least as fast as M7.

Some clarifications:

A path is simply connected if it does not contain any closed subpaths.

The 2xM rectangle encompass points [x,y] in the Cartesian plane with:

0 ≤ x ≤ 2 and 0 ≤ y ≤ M

Paths are constructed within the 2xM rectangle from vertical and horizontal segments connecting points with integral values in x and y.

An alternative statement of the problem is to show that the number of paths/M7 > 0 as M goes to infinity.

The boundary of the green area is just one such closed loop.
 y ↑M . ... ... . 2 1 0 1 2 x

 No Solution Yet Submitted by FrankM Rating: 3.0000 (1 votes)

 Subject Author Date re: That's puzzling (to Charlie) Charlie 2008-03-29 01:37:26 That's puzzling (to Charlie) FrankM 2008-03-28 21:28:58 re: A similar conclusion Charlie 2008-03-28 20:25:01 A similar conclusion Paul 2008-03-28 20:01:28 actual numbers Charlie 2008-03-28 14:50:35 Am I missing something? (spoiler?) Charlie 2008-03-28 11:15:12

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