So the condition would be that we have to transform the sequence $\{1,1,2,2,...,50,50\}$ into a sequence where there are $n$ elements between every two $n$... So between the 2 $1$'s there has to be $1$ element
Is there a sequence satisfying this condition?
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combinatorics
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0You're talking about sequences, not sets. – 2017-02-10
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0Well but is there one? – 2017-02-10
1 Answers
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This is Langford's problem. The number of solutions is given in OEIS A014552. There are no solutions for $n\equiv 1,2 \pmod 4$, so no solution with $n=50$. I found a solution $23421314$ for $4$ by hand then did a web search for that number. A sketch of the proof that there are no solutions for $n \equiv 1,2 \pmod 4$ is on this page under the heading Roy O. Davies finds key to the Solvability of 'n'