## 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)
Albert Stadler (20 December 20:29)
Bojan Bašić (23 December 23:22)
Rani Hod (26 December 23:39)

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!