Using your Head is Permitted

May 2008 riddle

Let n be an integer greater than 1, and let S_i for i between 1 and n-1 be a permutation over the numbers 1 through n-1. Define T_i as follows:

T₁=S₁ mod n
T₂=S₁+S₂ mod n
...
T_n-1=S₁+...+S_n-1 mod n

For which values of n does such a permutation, S_i, exist, where T_i is also a permutation over the numbers 1 through n-1?

For which values of n does such a permutation, S_i, exist, with the further constraint that for every i, S_i+S_n-i=n?

For credit, answer both questions with proof.

List of solvers:

Hongcheng Zhu (1 May 13:45)
Bojan Bašić (4 May 01:59)
Mark Tilford (4 May 11:39)
Dan Dima (7 May 05:25)
Or Sheffet and Itsik Horovitz (19 May 00:12)

Elegant solutions can be submitted to the puzzlemaster at riddlesbrand.scso.com. Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.

Enjoy!

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