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        riddles >> hard >> Rooks at integer distance
        
(Message started by: JocK on Aug 18^th, 2005, 12:46pm)

Title: Rooks at integer distance
Post by JocK on Aug 18^th, 2005, 12:46pm

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?

Title: Re: Rooks at integer distance
Post by Earendil on Aug 19^th, 2005, 7:18pm

Which definition of distance?

Title: Re: Rooks at integer distance
Post by JocK on Aug 20^th, 2005, 5:42am

on 08/19/05 at 19:18:16, Earendil wrote:

Which definition of distance?

Just the normal straight line distance.

(And of course both pieces are to be positioned exactly at the centres of their squares.)

Title: Re: Rooks at integer distance
Post by Icarus on Aug 20^th, 2005, 7:07am

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.)

Title: Re: Rooks at integer distance
Post by Earendil on Aug 20^th, 2005, 7:14pm

Which goes to 0, as proved on some topic somewhere on the hard forum

Title: Re: Rooks at integer distance
Post by JocK on Aug 21^st, 2005, 3:58pm

on 08/20/05 at 19:14:19, Earendil wrote:

Which goes to 0, as proved on some topic somewhere on the hard forum

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).

Title: Re: Rooks at integer distance
Post by Earendil on Aug 21^st, 2005, 6:04pm

on 08/21/05 at 15:58:24, JocK wrote:

Do you know the thread title? I am keen to have a look at it...

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 :'(

Title: Re: Rooks at integer distance
Post by JocK on Aug 22^nd, 2005, 4:51am

on 08/21/05 at 18:04:04, Earendil wrote:

[..] It was proved that the probability of choosing a hypothenuse number goes to 0 [..]

Actually, it was claimed in http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;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...)