The game Capture is played by two players, First and Second, who take turns to move towards one another on a narrow bridge. Initially, the two players are at opposite ends of the bridge, at a distance of n feet from each other. (Here n is an arbitrary positive integer.) When it is their turn, they are allowed to jump 1, 2, or 3 feet toward each other. The game starts by First making a move first;
The game Capture is played by two players, First and Second, who take turns
to move towards one another on a narrow bridge. Initially, the two players are
at opposite ends of the bridge, at a distance of n feet from each other. (Here n
is an arbitrary positive integer.) When it is their turn, they are allowed to jump
1, 2, or 3 feet toward each other. The game starts by First making a move first;
the game is over when one player is able to capture (jump on top of) the other
player.
(a) Which player has a strategy to capture the other player if n= 8?
(b) What if the initial distance is n= 21 feet?
(c) Generalize the problem for all positive integers n.
(d) Generalize the problem further to the case when the initial distance is n feet
and the players are allowed to jump any (positive) integer number of feet
up to k feet. (Here n and k are arbitrary natural numbers.)
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