MathDB

Let

k

n

be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers

1, 2 \dots, n

, with

1

and

n

neighbors. It is known that pin

1

is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer

d

any and Bernaldo moves the ring

d

pins clockwise or counterclockwise (positions are considered modulo

n

, i.e., pins

x

y

equal if and only if

n

divides

x-y

). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:
Rule 1: Arnaldo chooses a positive integer

d

and Bernaldo moves the ring

d

pins clockwise or counterclockwise.
Rule 2: Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in

d

kd

pins, where

d

is the size of the last displacement performed.
Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of

k

, the values of

n

for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.

3

Problems(2)