Ray Kremer
Further correct solutions came from Jens Voss, Philippe Fondanaiche,
Michael Lynch.
There are 42 creek meanderings with seven crossings.
Imagine the road crossing over the creek on seven bridges numbered consecutively from 1 to 7. To each meandering we can assign a permutation of the integers 1 through 7 by writing down the order in which the creek meets the bridges. In the examples with five crossings given in the statement of the problem, the top four meanderings would be labeled 12345, 12543, 14325, and 32145. Note that you can get the bottom four meanderings by reflecting the first four about the center bridge.
But not all such labelings correspond to a possible meandering. For example, 1235467 is impossible -- sketch it and see! A little thought will convince you that crossings from the north must take place at an odd numbered bridge while those from the south take place at an even numbered bridge. This means that any acceptable labeling must start with an odd number and alternate odd and even numbers. Since there are three even numbered bridges and four odd numbered bridges, this gives an upper bound of 4!3! = 144 meanderings. By the reflection principle, we can assume that the first two odd numbers are 1 or 3 which reduces the possibility in half to 72.
These can be quickly checked by hand to yield the 42 crossings.
There appears to be no general formula known for the
number of meanderings with k crossings. Thanks to Jens Voss for the
reference to the web site
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Page last updated 2 April 2001.
ã 2001 Alberto L Delgado