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Dao Juxing (The way of the rectangle) (Posted on 2008-03-28) Difficulty: 4 of 5

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)

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: That's puzzling (to Charlie)Charlie2008-03-29 01:37:26
Some ThoughtsThat's puzzling (to Charlie)FrankM2008-03-28 21:28:58
re: A similar conclusionCharlie2008-03-28 20:25:01
A similar conclusionPaul2008-03-28 20:01:28
actual numbersCharlie2008-03-28 14:50:35
SolutionAm I missing something? (spoiler?)Charlie2008-03-28 11:15:12
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