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Chess 2: The Sequel» Forums » General

Subject: 21 not 36 match ups? rss

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Chris Rudram
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With six sides available to each player, shouldn't that make 6 x 6 potential match ups? Not quite sure where 21 comes from...
 
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David Sirlin
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The number of matchups for a game with 6 sides is (6 choose 2) + 6. That's 15 + 6 = 21.

6 choose 2 is a combination (as opposed to a permutation) that gives the number of different ways you can choose 2 things from a set of 6 where order doesn't matter. Order not mattering means that the matchup of A vs B should count the same as B vs A. The + 6 at the end is for the 6 mirror matchups. You can read about combinations here: http://en.wikipedia.org/wiki/Combination

6x6 double counts a lot of matchups. It's counting A vs B and B vs A as different matchups when really that is the same matchup.
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David desJardins
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By the way, the number of ways of choosing a set of K objects of N different types, where the order of the elements in the set doesn't matter, is (N+K-1 choose k). So this is a special case where K=2 and N=6 and so the number of sets is (7 choose 2) = 21.
 
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Chris Rudram
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Okay I was counting White Reaper vs. Black Animal as a different match up to White Animal vs. Black Reaper, assuming order was important.

That clears that up, thanks.

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David Sirlin
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Oh...yes you could count that way. It's true that White Reaper vs. Black Animal and White Animal vs. Black Reaper are slightly different. (Just like white and black are slightly different in chess.) It's just that difference is very small *compared* to the difference between whole armies. In an asymmetric game, the term "matchups" usually applies to these big differences such as Animal vs Reaper and Animal vs Empowered counting as different, but not Reaper vs Animal and Empowered vs Animal counted again. (Even though yes there is some difference in who goes first). I see what you meant though now.
 
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