## December 2007 riddle

Let n>1 be an integer, let S be the set of numbers {0, 1, ... , n-1} and let A, B and C be one-to-one functions to and from S.

#### Part 1: The Addition Problem

For which values of n is it possible to find such A, B and C satisfying that for all x in S, A(x)+B(x)=C(x) (mod n)?

#### Part 2: The Multiplication Problem

For which values of n is it possible to find such A, B and C satisfying that for all x in S, A(x)*B(x)=C(x) (mod n)?

#### Bonus Question: The Power Problem

For which values of n is it possible to find such A, B and C satisfying that for all x in S, A(x)B(x)=C(x) (mod n)?

Note for the last part that 00 is not defined.

In order to be considered a solver for any of the parts, you should provide for each n either an example for such A, B and C or proof that such A, B and C do not exist.

A separate list of solvers will be kept for each of the two parts. Answer either or both.

### List of solvers:

#### Part 1:

Ross Millikan (6 December 04:17)
David Jager (6 December 23:20)
Wolfgang Kais (9 December 23:15)
Dan Dima (14 December 08:11)
Rick Kaye (14 December 18:04)
Ananda Raidu (31 December 20:34)
Mark Tilford (1 January 01:43)

#### Both parts:

Omer Angel (1 December 05:25)
Oded Margalit (17 December 11:16)
Itsik Horovitz (18 December 20:59)
Hongcheng Zhu (19 December 22:54)
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