# Can the solution a) of the accepted answer to the "Camels crossing a bridge" problem be extended to arbitrarily many camels?

This problem (and solution) with 5 camels each was posted here on Puzzling several years ago:

Two merchants have 5 camels each. One fine day, one of the merchants sells all of his goods in the market and heads back to the village. He has to cross a bridge to reach his destination, and the bridge is only wide enough for 1 camel at a time.

At the same time, the other merchant is heading to the market to sell his goods and must cross the same bridge. Alas! They meet up in the middle of the bridge one camel-length from each other. They are both stubborn and won't back up their camels. Fortunately, the first merchant's camels are spry and unladen and are able to jump over another camel. Can both merchants get their camels across the bridge? (Both merchants' camels can jump)

The accepted answer offers two solutions, the first of which - 'a)' - goes as follows:

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Question: Can this solution be extended to arbitrarily many camels?

If either kind of camel can jump, then yes:

A*(n+2) _ (AB)*i B*(n+2) (split off 2As from the first group)
A*n A A _ (AB)*i B B B*n (the leftmost Bs jump over an A)
A*n A A (BA)*i _ B B B*n (one B advances a step)
A*n A A (BA)*i B _ B B*n (The rightmost As jump over a B)
A*n A _ (BA)*i B A B B*n (Join a BA to the middle group)
A*n A _ (BA)*(i+1) B B*n (one A advances)
A*n _ A (BA)*(i+1) B B*n (join an A and a B to the middle group)
A*n _ (AB)*(i+2) B*n

Keep following this until you run out of As.

If n is even, at this point the camels will be alternating. Bs can just jump over As until they are off the bridge, then the As can just walk forward.

If n is odd, then all the Bs but the rightmost jump forward once. The rightmost B moves forward once. Now camels are alternating. Bs can jump forward until all are off the bridge.