Find the Next No.

Look at this sequence from top to bottom. What is the next number in the sequence?
1
11
21
1211
111221
312211




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Now P takes one minute to cross the bridge, Q takes two minutes to cross, R takes five minutes to cross and S takes ten minutes to cross.

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No Solution

In 2007, a puzzle was released and $2 million prizes were offered for the first complete solution. The competition ended at noon on 31 December 2010, with no solution being found. Wiki