The general problem is to find n+1 permutations of the sequence 1,2,3,...,2n such that all (n+1)(2n-1) partial sums are distinct, or equivalently, such that every integer strictly between 0 and 1+2+3+...+2n occurs exactly once as a partial sum. (Here a partial sum of a sequence is neither allowed to be empty nor allowed to coincide with the original sequence.) E.g., for n=2, we can use 1,2,3,4 (with partial sums 1,3,6) and 4,3,1,2 (with partial sums 4,7,8) and 2,3,4,1 (with partial sums 2,5,9). It seems that this can be done for all n, but nobody knows how to prove it. Jim On Friday, January 8, 2016, Dan Asimov <asimov@msri.org> wrote:
On Jan 8, 2016, at 6:22 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote: . . . . . . I just learned about another nice example from Barry Cipra. His Bricklayer's Challenge ( http://www.pavelspuzzles.com/2012/11/the_bricklayers_challenge.html) appears to be solvable for every n, but nobody knows how to prove it for infinitely many n.
At that URL I see three challenges mentioned, none of them depending on an integer n.
Feel free to clarify.
—Dan
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