Correct solutions were submitted by Steve Depies, Ray Kremer, Adam Florzak, Kim Heintz, Eric Stein, Shreeti Shrestha, Jason Doll, Rich Bernstein, Brian Papciak, Mundia Mubyana, Nicole Fruitman.  Further correct solutions came from Roger Stites, Sudipta Das, Michael Lynch, Tom Kauffman, Robert McQuaid, Ivan Lisac, Stepehen Smith, Philippe Fondanaiche, Edward Lee, Steve Prowse, Francesc Suñol.

There are 12,870 ways.

Two methods of solution present themselves.

In the first, you observe that the number of paths going through any one given letter in the array is the sum of the number of paths that go through the two letters directly above it (there is only one such letter if it's on an end), and that there is only one path going through a letter on the edge.  This is precisely the defining property of the binomial coefficients.  So the number of paths is the number located in the middle of the 17th row of Pascal's triangle, which is the number above.

In the second, notice that each path from top to bottom must make eight turns to the southEast and eight turns to the southWest -- there are clearly sixteen turns and, as you end up directly below where you started, the number of southEast turns must equal the number of southWest turns.  So, on the one hand, every path can be written as a sixteen letter word spelled with exactly eight E's and eight W's, while, one the other hand, every such word corresponds to exactly one path.  In order to write such a word, we need only choose which of the sixteen letters is an E, which gives the binomial coefficient C(16,8), again the answer above.

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Page last updated 2 April 2001.