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Topic: Catch the thief (Read 630 times) |
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Aryabhatta
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Two people play a game on the board given below. One guy plays as the thief (marked in red) and other plays as the cops (two guys in blue). The initial positions are as shown in the figure. The players take turn moving their pieces one step at a time. The thief moves first. Then both the cops have to move one step each. (Everyone *has* to move, no pass...) and so on. The goal of the second player is to catch the thief.
The thief is caught if:
RuleSet 1) The thief cannot move on his turn without landing on a cop OR a cop lands on the thief on his move.
RuleSet 2) The cops move during the day (and sleep at night) and the thief moves during the night (and sleeps during the day), so the thief can land on a cop during his move. The thief is caught only if the cop lands on the thief on his move.
Who wins with RuleSet 1 and who wins with RuleSet 2?
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Icarus
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Re: Catch the thief
« Reply #1 on: Oct 21st, 2004, 7:23pm » |
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As you have stated the rules, clearly either player 2 wins or it is a stalemate. Why? Because player 1 has no winning condition by either rule set!
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Aryabhatta
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Re: Catch the thief
« Reply #2 on: Oct 21st, 2004, 8:25pm » |
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on Oct 21st, 2004, 7:23pm, Icarus wrote:| As you have stated the rules, clearly either player 2 wins or it is a stalemate. Why? Because player 1 has no winning condition by either rule set! |
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You got me there 
OK. How about: Player 1 wins if a position repeats.
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rmsgrey
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Re: Catch the thief
« Reply #3 on: Oct 22nd, 2004, 2:50am » |
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An alternate victory condition for the thief is to prolong the game indefinitely - in other words, to prove that the cops can never catch him from the current position. Another way of phrasing it is to say that in the case of a stalemate, the thief wins.
Ruleset 1:
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Cops win since the left cop can remain one move above and to the left of the thief while the other cop loops round the edge in a clockwise motion. (OK, I'm not great at description...)
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Ruleset 2:
Hint (bit of a giveaway)::parity::
Solution
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Thief wins (can prolong the game indefinitely) since all loops take an even number of steps to complete, so you can colour the nodes with 2 colours so that no two nodes of the same colour share an edge - eg red and blue - since the thief starts on a red node, and the cops on blue nodes, each time the thief moves, he moves onto the same colour as the cops current colour, and each time the cops move, they move onto the other colour from the thief, so can never make a capturing move (which would be a move onto the same colour as the thief). In fact, no matter how badly the thief plays, he can never be caught - it's like playing checkers with one player's pieces on the white squares and the other's on the black squares...
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[e]Broken tag...[/e]
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| « Last Edit: Oct 22nd, 2004, 2:51am by rmsgrey » |
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Aryabhatta
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Re: Catch the thief
« Reply #4 on: Oct 22nd, 2004, 2:44pm » |
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Right rmsgrey! It was too easy for you, wasn't it? 
It is really tough to come up with medium puzzles!
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