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Topic: King Arthur (Read 708 times) |
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FiBsTeR
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Sorry if my English isn't gangster-ized...
Anyway, this was a warm-up problem I had in my discrete math class today:
King Arthur and his twelve knights are to be seated around the Round Table, but they lost the seating arrangement chart. Two seating arrangements are the same iff each person is sitting next to the same two people in both arrangements.
a) How many different seating arrangements are there?
b) If King Arthur has a special seat to himself, how many different seating arrangements are there?
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| « Last Edit: Sep 5th, 2007, 6:56pm by FiBsTeR » |
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Sameer
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Re: King Arthur
« Reply #1 on: Sep 5th, 2007, 7:23pm » |
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1) Its a free circular permutation (n-1)!/2. In this case 12!/2
My permutation is little weak
2) I think if we fixed 1 seat, so it will be 11!/2. Not sure about it though!!
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| « Last Edit: Sep 5th, 2007, 7:25pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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SMQ
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Re: King Arthur
« Reply #2 on: Sep 6th, 2007, 8:16am » |
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Are mirrored arrangements considered the same? Each person is sitting next to the same people, but on opposite sides.
If not: 1) total permutations of seats divided by number of seats (because rotations around the table are the same) = 13!/13 = 12! 2) total number of permutations of knight's seats (rotations are no longer the same) = 12!. If mirrored arrangements are the same, divide each of the above by two to account for the symmetry. In either case there's no difference whether or not King Arthur's seat is fixed!
--SMQ
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rmsgrey
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Re: King Arthur
« Reply #3 on: Sep 6th, 2007, 9:37am » |
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on Sep 6th, 2007, 8:16am, SMQ wrote:| In either case there's no difference whether or not King Arthur's seat is fixed! |
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Which is unsurprising if you think about it - when Arthur's seat isn't fixed, you can always rotate the table so he's in the same place without changing to a different seating arrangement.
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FiBsTeR
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Re: King Arthur
« Reply #4 on: Sep 6th, 2007, 3:05pm » |
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Your second part is correct, SMQ; mirrored arrangements are considered the same by the definition I gave in the problem.
If anyone has any problems seeing why the answer does not change when "restricting" the King's seat, notice that you can think of the second question (b) as lining up the knights after the King's seat; the problem thereby loses its "circular" properties.
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