In year 6002 planet Earth has reached a high degree of civilization
and war has essentially disappeared. Only a vestige of war survives in
the way international controversies between countries are resolved:
The two opponent countries, say C1 and C2, set up a square
battlefield and they turn in placing their tanks in the field. Tanks
cannot overlap. The winner is the last country which is able to place
a tank.
To make the process more disciplined, the battlefield is divided into
n x n squares (as a kind of chessboard), where n is even, and
each tank occupies two adjacent squares, horizontally in a row or
vertically in a column.
As an exemplification, you can see below a possible configuration of
the field after four turns. The tanks placed by C1 are green, while
those of C2 are red.
The battle is normally very quiet ... the two opponent countries
are quite nervous only when they have to decide who should start
placing the first tank.
Q1. Can you explain why? If C1 is the country that
starts placing the tanks, does winning strategy for C1 or for
C2 exist?
Answ.: Yes, there is a winninig strategy for C2. If C1 places
a tank at a certain position, say p, C2 always answer by placing a
tank at a position p' which is symmetric to
p with respect to the
center of the field. Since this maintains the symmetry of the field,
whenever C1 is able to place a tank, also C2 will be able to do so,
and thus C2 will surely be the winner.
A possible configuration of the field after four steps is exemplified
below. Tanks are numbered as x.y where x stands for the owner and
y is a progressive number denoting at which turn the tank has been
placed.
Note that the above argument applies to the case of "ordinary" tanks,
which need a free path to access there final position, and also to
flying tanks, which can be placed in any free position in the field,
even if it not accessible.
Once the existence of a winning strategy is discovered, the regulation
is drastically changed: the battlefield is no more organised in a
grid, and now the tanks can be placed in any position, with any
orientation, but still avoiding any overlapping.
Q2.Do you think that this improves the situation?
Answ.: Not really. Now there is a winning strategy for C1. In
fact, C1 can start placing the tank at the center of the board. Then,
the previously illustrated strategy can be reused: at any turn C1 will
play symmetrically with respect to the move of C2 and thus C1 will be
the winner.
A possible configuration of the field, after five moves of C1 and
four moves of C2 can be found below.
