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ideas which can be used to unify treatments for a variety of games, in other words,
usable methods. Thus the choice has been strongly influenced by two factors. Firstly
there needs to be a connection between the games and secondly each game has to
provide a stimulus for further research; in most cases this means that the games are
used as a starting off point from which attractive open questions can be generated.
As a result Sects. 6.2 and 6.8 all include suggestions for further work of varying
degrees of difficulty which are intended to encourage more researchers to take an
interest in the games. However it does mean that other games associated with Ruckle
such as lattice and accumulation games have been ignored. In keeping with the spirit
of Ruckle's topic, the games in this chapter are all two-person zero-sum ones and
the players are called RED and BLUE with RED being the maximizer. The structure
of the chapter is described in the following paragraphs.
The Several Intervals Game is played in the unit interval I with BLUE choosing
a point of I and RED simultaneously selecting intervals of given lengths
i in I ;the
payoff to RED is one if BLUE's point is in one of the intervals and zero otherwise.
Although simply stated, this has proved to be an extremely difficult game to solve
and no comprehensive solution has been found when RED can choose more than
two intervals. Abbreviated details of the original (unpublished) approach used for
the Two Intervals Game and some of the results on the Three Intervals case are given
in Sect. 6.2 .
Ruckle's greedy games have not attracted very much attention and are the subject
of Sects. 6.3 and 6.4 . Many games have the form that RED has the task of deciding
where to hide a given amount of material but, in greedy games, RED has the addi-
tional decision of determining how much material to hide when facing the prospect
that hiding more means a greater probability of discovery. Section 6.3 gives a for-
mal definition of a greedy game and then concentrates attention on games in the unit
interval whereas Sect. 6.4 looks briefly at greedy games on the unit square.
In the Number Hides Game RED and BLUE simultaneously choose subintervals
of given length in an integer interval with RED getting a payoff equal to the num-
ber of integers the subintervals have in common. The game proved more tractable
than the other games we consider and it was solved independently by three sets of
researchers. Section 6.5 presents some natural variations of the game and Sect. 6.6
discusses the interesting generalization by Zoroa, Fernandez-Saez and Zoroa [ 7 ]in
which BLUE has to hide a given quantity of objects in an integer subinterval of his
choice with the stipulation that at least one object and at most c can be placed at
each integer of the subinterval; as before RED chooses a subinterval and receives
an amount equal to the number of objects in it. The solution of this game seems to
be difficult so, as a first step, it is proposed that the solution of a couple of particular
cases extending those of [ 7 ] be attempted.
The Hiding in a Disc Game is again easily stated. RED and BLUE simultane-
ously choose points in the unit disc and RED wins if and only the chosen points are
at most a given distance c apart. It was already proving awkward 30 years ago as
Ruckle demonstrated that an assertion concerning its value in the American Mathe-
matical Monthly was false and, since then, there seems to have been little work done
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