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Topic: Rooks at integer distance (Read 1310 times) |
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JocK
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Rooks at integer distance
« on: Aug 18th, 2005, 12:46pm » |
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Randomly place a white and a black rook on a large chessboard consisting of N x N unit squares in such a way that both rooks can not capture each other.
For large N, what is the likelihood that both rooks are at integer mutual distance?
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Earendil
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Re: Rooks at integer distance
« Reply #1 on: Aug 19th, 2005, 7:18pm » |
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Which definition of distance?
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JocK
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Re: Rooks at integer distance
« Reply #2 on: Aug 20th, 2005, 5:42am » |
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on Aug 19th, 2005, 7:18pm, Earendil wrote:| Which definition of distance? |
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Just the normal straight line distance.
(And of course both pieces are to be positioned exactly at the centres of their squares.)
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Icarus
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Re: Rooks at integer distance
« Reply #3 on: Aug 20th, 2005, 7:07am » |
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For large N, this should approach the asymptotic density of non-zero pythagorean pairs among all integer pairs. ( (x, y) is a pythagorean pair if x2 + y2 is a perfect square.)
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Earendil
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Re: Rooks at integer distance
« Reply #4 on: Aug 20th, 2005, 7:14pm » |
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Which goes to 0, as proved on some topic somewhere on the hard forum
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JocK
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Re: Rooks at integer distance
« Reply #5 on: Aug 21st, 2005, 3:58pm » |
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on Aug 20th, 2005, 7:14pm, Earendil wrote:| Which goes to 0, as proved on some topic somewhere on the hard forum |
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Do you know the thread title? I am keen to have a look at it...
And I am interested in how this density behaves asymptotically as function of N (with the answer to this problem one should be able to calculate the probability asked for with a relative uncertainty that drops to zero for N --> Infty).
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Earendil
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Posts: 46
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Re: Rooks at integer distance
« Reply #6 on: Aug 21st, 2005, 6:04pm » |
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on Aug 21st, 2005, 3:58pm, JocK wrote:
Do you know the thread title? I am keen to have a look at it...
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Ops... my mistake. It was proved that the probability of choosing a hypothenuse number goes to 0, not the probability of a pitagorean pair. sorry 
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JocK
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Re: Rooks at integer distance
« Reply #7 on: Aug 22nd, 2005, 4:51am » |
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on Aug 21st, 2005, 6:04pm, Earendil wrote:
[..] It was proved that the probability of choosing a hypothenuse number goes to 0 [..]
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Actually, it was claimed in http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_har d;action=display;num=1124036799 that for very large integers the probability of these not being a hypothenuse number approaches to zero.
(An asymptotic expression for this probability has not been posted yet...)
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| « Last Edit: Aug 22nd, 2005, 4:57am by JocK » |
IP Logged |
solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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