general notion formally although we lay out complete specifications of the
strategies we report. We will use uppercase letters S, T , X , Y to denote
pure strategies. Strategy X is called a winning strategy (a tying strategy)
for player p if no matter what player q does, by using X player p guarantees
that Sp > Sq (Sp ≥ Sq). Throughout the paper we will use the standard
notation m | n, to denote the fact that m divides n and m - n, to denote its
negation.
2.1 Some definitions and existing results
We first develop some concepts and notations. Let C be a circle and let
P ⊆ C be a finite set of points on the circle. Then an arc (x, y) ⊆ C
such that {x, y} ⊆ P and (x, y) ∩ P = 0 is called an interval. Now let
PR and PG such that PR ∪ PG = P be sets of red and green points of P ,
respectively. Then an arc (x, y) ⊆ C such that {x, y} ⊆ PR ({x, y} ⊆ PG)
and and (x, y) ∩ PR = 0 ((x, y) ∩ PG = 0) is called a red (green) interval.
An interval that is neither red nor green is called a bichromatic interval
and an interval which is not bichromatic shall be at times referred to as
a monochromatic interval in general. We will use rC (gC) to denote the
number of red (green) points placed on the circle C. We will also use IR(C)
(IG(C)) to denote the number of red (green) intervals on the circle C.
Given a circle C and a point x ∈ C , an antipode of x is the point y ∈ C
such that d(x, y) = 1/2. The pair of points {x, y} is called a pair of antipodes
of C. Let m be a positive natural number. Then the set of key positions 5
on C determined by point x and m is the set
κ(C, x, m) = {p ∈ C : a(p, x) = l/m, where l ∈ {0, . . . , m - 1}}.
By the set of key positions determined by m we mean a set of key positions
5We use a term key position here for what was called a key point in the paper Ahn
et al. [2004]. We found the name key position somewhat more apropriate in our context.