# Self-power number sequences

N. J. A. Sloane njas at research.att.com
Sun Feb 3 02:39:20 CET 2008

>From: "Bob Hearn" <seqfan at hearn.to>
> Not specifically, but in a statistics class at Rice we once computed  the
> expected number of moves required by a knight on a central square  of a
> chessboard to return to its starting square, making legal moves  at
> random. Surprise: the answer is 42. One could generalize that  problem as
> well, but the Tower of Hanoi problem seems much more natural.
>

Experimentation  (10 000 000 tours simulation, for each N <= 12 ) seems to
show that for all boards NxN (N >=5) ,
the expected number of moves is :

=  (N-1)x(N-2) for a knight on a central square (that is 42 (!) for N=8)
=  4x(N-1)x(N-2) for a knight on a corner square
=  a multiple of N-1 or N-2 for a knight on any square

Anybody has a proof  (disproof) of it ?

Regards,
JT

http://www.echolalie.com
http://www.echolalie.org/wiki

It was off the air for about 60 hours because
of computer (and support) problems.

My apologies

Neil Sloane

Very interesting; I would not have expected a simple pattern. I do
remember (this was 1984?, so I am reaching back here) that we solved
the problem analytically with Markov chains. So the answer can be
gotten exactly without simulation.

Bob

On Feb 4, 2008, at 8:03 AM, Jacques Tramu wrote:

>
>> From: "Bob Hearn" <seqfan at hearn.to>
>> Not specifically, but in a statistics class at Rice we once
>> computed  the expected number of moves required by a knight on a
>> central square  of a chessboard to return to its starting square,
>> making legal moves  at random. Surprise: the answer is 42. One
>> could generalize that  problem as well, but the Tower of Hanoi
>> problem seems much more natural.
>>
>
> Experimentation  (10 000 000 tours simulation, for each N <= 12 )
> seems to show that for all boards NxN (N >=5) ,
> the expected number of moves is :
>
> =  (N-1)x(N-2) for a knight on a central square (that is 42 (!) for
> N=8)
> =  4x(N-1)x(N-2) for a knight on a corner square
> =  a multiple of N-1 or N-2 for a knight on any square
>
> Anybody has a proof  (disproof) of it ?
>
> Regards,
> JT
>
> http://www.echolalie.com
> http://www.echolalie.org/wiki