## Ken's Puzzle of the WeekCrossing a River

A river runs East to West and is 30 feet wide.  In the river are six posts, three in an East-West line, 10 feet from the southern shore, and three in an East-West line, 10 feet from the northern shore.  The posts are 10 feet apart.  10-foot planks can be attached between the posts, or between a post and the shore.  With enough planks there are various ways to create a bridge from North to South.

The posts are only 1 foot above the river.  A person in a raft wishing to travel from East to West will be blocked by any planks.  If the planks create a complete bridge, there is no route for the raft.

Which is larger: the number of different routes for a walker from North to South, or the number of different routes for a raft from East to West? How many different routes for each are there?

Source: Past puzzle experience.
Solutions were received from Kirk Bresniker, Jean Moreau de Saint-Martin, and Yaacov Yoseph Weiss.

From North to South, there are:
a) 3 options from the northern shore to the first row of posts
b) 3 options from the first row of posts to the second row of posts
c) 3 options from the second row of posts to the southern shore

In addition there are two paths that progress south to the second row, return to the first row, then proceed to the southern shore.

Altogether, there are 29 different paths from North to South.

The exact same analysis can be made for East-West travel on the river, also leading to 29 routes.

A similar problem  (problem G129) was proposed resently on Philippe Fondanaiche's website (in French) www.diophante.fr.

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