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Let $k < n$, $k$ even, $n$ odd. Mary is to play $n$ chess games against her parents, alternating between her father and mother. To receive her allowance she must win $k$ games in a row. Given the choice, should she start against the stronger or weaker parent?
The puzzle source is apparently Peter Winkler.
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I’m going to go counter-intuitive and say…
Mary should play the stronger parent first, as that will give her more opportunities to win against them.
In a simplified example, say N = 5 and K = 4. Mary has to win games 2-4, and then either game 1 or game 5. If she plays the stronger parent first, she will have to win the WSW series in the middle (where W = Weaker and S = Stronger) and then can win either the 1st or last game against S… giving her two chances. This way she only has to win 2/3 games against S, and 2/2 games against W which sounds way easier to me.
In a set where N = 9 and K = 4, if she starts with the stronger parent, again she’d have more chances to beat them. She would need to win at least 2/4 (50%) against the weaker parent, but only 2/5 (40%) against the stronger one.
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I think that Mary should first play against
the weaker parent.
Because $k$ is an even number, Mary would have to win $k / 2$ games against each parent. Since it would be better for Mary to win sooner rather than later, this strategy would give her a greater chance at winning the initial game, allowing her to continue her streak.
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