Knight's Tour: Are there other similar problems?
Knight's Tour: Are there other similar problems?
Hi (from a newbie)
I have recently published an article about Knight's Tours (see https://theconversation.com/howtoget ... blem22282).
It got me thinking whether there are any other problems like this, as I'd like to look at doing some research into variants of Knight's Tours, or other Chess related problems.
Does anybody know of any?
Thx
G
I have recently published an article about Knight's Tours (see https://theconversation.com/howtoget ... blem22282).
It got me thinking whether there are any other problems like this, as I'd like to look at doing some research into variants of Knight's Tours, or other Chess related problems.
Does anybody know of any?
Thx
G

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
Go to 
http://www.chessbase.com
Then enter the word puzzles in the search box (top right).
You may find some chess puzzles of interest to you that way.
http://www.chessbase.com
Then enter the word puzzles in the search box (top right).
You may find some chess puzzles of interest to you that way.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)
Re: Knight's Tour: Are there other similar problems?
Many thanks  that is very useful.
G
G

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
From reading "theconversation" you link to in your original post I see why you are interested in knight tours.
The Oxford Companion to Chess 1984 has 
"There is almost an infinity of ways of achieving this (knight's tour) and more than 122,000,000 ways of performing the more restricted version known as the reentrant tour...
Mathematicians have derived formulae for generating tours and attention... directed to particular versions such as symmetrical tours..."
A pure mathematician would balk at "almost an infinity of ways" and perhaps "have derived formulae".
And, in your "theconvsation" there is wriiten, "Darwin's principle of natural selection  survival of the fittest."
A politicallycorrect geneticist might well take exception to that.
In number theory the hard task is "deriving a formula" (algorithm?) that will find all the prime numbers, or proving by mathematical logic that there is an infinity of them.
In chess the hard task is to find the number of checkmate positions that exist in the 'almost infinite' number of chess positions.
The Oxford Companion to Chess 1984 has 
"There is almost an infinity of ways of achieving this (knight's tour) and more than 122,000,000 ways of performing the more restricted version known as the reentrant tour...
Mathematicians have derived formulae for generating tours and attention... directed to particular versions such as symmetrical tours..."
A pure mathematician would balk at "almost an infinity of ways" and perhaps "have derived formulae".
And, in your "theconvsation" there is wriiten, "Darwin's principle of natural selection  survival of the fittest."
A politicallycorrect geneticist might well take exception to that.
In number theory the hard task is "deriving a formula" (algorithm?) that will find all the prime numbers, or proving by mathematical logic that there is an infinity of them.
In chess the hard task is to find the number of checkmate positions that exist in the 'almost infinite' number of chess positions.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)
Re: Knight's Tour: Are there other similar problems?
John
Thanks. I realise that the article would not pass strict scientific rigour but the article is written for a slightly different audience and I hope that the terminology does not upset too many people.
Thanks. I realise that the article would not pass strict scientific rigour but the article is written for a slightly different audience and I hope that the terminology does not upset too many people.

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
Thanks for your explanation and for your link, which I read with interest.
I am known as a bit of a jester or a complete fool by some on this forum so no need to take me totally seriously, dude
Watch out here, though, too many are too easily upset if you touch on one of their many nerves. I, of course, just get on their nerves, constantly.
I am known as a bit of a jester or a complete fool by some on this forum so no need to take me totally seriously, dude
Watch out here, though, too many are too easily upset if you touch on one of their many nerves. I, of course, just get on their nerves, constantly.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)
Re: Knight's Tour: Are there other similar problems?
John
Thanks again  appreciate it.
G
Thanks again  appreciate it.
G

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
Nada. Hasta luego, amigo.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
I forgot to ask if Professor Kendall could say what the next hard task in his own field was or if anyone knew what it was in genetics? Never mind  everything comes to he who waits.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)

 Posts: 1347
 Joined: Fri Jun 01, 2007 6:31 pm
Re: Knight's Tour: Are there other similar problems?
Euclid would not be happy you dismiss his proof of 300 BC !John McKenna wrote:...... the prime numbers, or proving by mathematical logic that there is an infinity of them.
There is a 3D knight's tour on the internet.
 Rob Thompson
 Posts: 757
 Joined: Wed Jul 15, 2009 12:03 pm
 Location: Behind you
Re: Knight's Tour: Are there other similar problems?
https://www.amazon.co.uk/AcrossBoardM ... 0691154988 is a book detailing knights tourns on n*m boards, as well of boards with different topologies (spheres, toroids, Klein bottles etc.). I recall the maths being not entirely trivial as presented, but that was a few years ago and I've developed considerably as a mathematician since then so that may not be an accurate perception.
True glory lies in doing what deserves to be written; in writing what deserves to be read.
 Christopher Kreuzer
 Posts: 7472
 Joined: Fri Aug 06, 2010 2:34 am
 Location: London
Re: Knight's Tour: Are there other similar problems?
Speaking of chess problems and mathematics, I've looking at some proof games over the past few weeks (these are positions that are reached from the starting position in a single, unique set of moves) and got to wondering how that sort of thing would be expressed mathematically.
PS. Why is this thread in 'Not Chess!'?
PS. Why is this thread in 'Not Chess!'?

 Posts: 3883
 Joined: Tue May 17, 2011 2:02 pm
Re: Knight's Tour: Are there other similar problems?
Salutem Plurinam Dicit to E Michael White and Rob Thompson.
Thanks to both for the N's tour info.
EMW>Euclid would not be happy you dismiss his proof...<
I am not the only one.
For it is a proof NOT universally acknowledged... for it must be in want of a contradiction.
In 1923 L. E. J. Brouwer 'dismissed' such proofs as unsound and proposed a different basis for mathematical logic.
Not a position that has ever been very popular but one that has not been refuted.
At least not to my knowledge that is. Being a bit of a maverick I quite like his ideas.
Christopher K>Speaking of chess problems and math...
PS Why is this thread in 'Not Chess'?<
Hi Chris, to answer the postscript  Professor Kendall probably does not know the ropes here never mind the threads.
Thanks to both for the N's tour info.
EMW>Euclid would not be happy you dismiss his proof...<
I am not the only one.
For it is a proof NOT universally acknowledged... for it must be in want of a contradiction.
In 1923 L. E. J. Brouwer 'dismissed' such proofs as unsound and proposed a different basis for mathematical logic.
Not a position that has ever been very popular but one that has not been refuted.
At least not to my knowledge that is. Being a bit of a maverick I quite like his ideas.
Christopher K>Speaking of chess problems and math...
PS Why is this thread in 'Not Chess'?<
Hi Chris, to answer the postscript  Professor Kendall probably does not know the ropes here never mind the threads.
To find a for(u)m that accommodates the mess, that is the task of the artist now. (Samuel Beckett)

 Posts: 152
 Joined: Tue Dec 22, 2009 4:14 pm
 Location: South Shields
Re: Knight's Tour: Are there other similar problems?
Back in my schoolteaching days I used to publish a regular puzzle the school newsapaper. Several of these involved a little chess
(a) How can you place 8 queens on a chessboard so that none of them can capture another?
(b) What is the maximum number of Rs, Ns, Bs that can be placed likewise so that none can capture each other?
(c) How can you place 5 Qs on a board so that all squares are covered by at least one of them (Two completely distinct solutions)
(d) In question (a) one cleverclogs claimed the answer was 64 as they were all white Qs and white pieces can't capture white Qs! He also produced an elegant position with 8 white Qs and 8 black Qs so that no captures could be made. Using this as a basis it is not difficult to place 9 Qs of each colour on the board without captures being possible and, I seem to remember that we managed 10Qs of each colour being the best we could do  but i can't remember the solution
(e) Various Martin Gardner 'Scientific American' puzzles involving chess boards, one of which was what was the maximum number of squares that could be coverd on a board with just one of each piece K,Q,R,B.N and P? These also included the old "proof" that 64 = 65 and multiple problems involving polyominoes
(a) How can you place 8 queens on a chessboard so that none of them can capture another?
(b) What is the maximum number of Rs, Ns, Bs that can be placed likewise so that none can capture each other?
(c) How can you place 5 Qs on a board so that all squares are covered by at least one of them (Two completely distinct solutions)
(d) In question (a) one cleverclogs claimed the answer was 64 as they were all white Qs and white pieces can't capture white Qs! He also produced an elegant position with 8 white Qs and 8 black Qs so that no captures could be made. Using this as a basis it is not difficult to place 9 Qs of each colour on the board without captures being possible and, I seem to remember that we managed 10Qs of each colour being the best we could do  but i can't remember the solution
(e) Various Martin Gardner 'Scientific American' puzzles involving chess boards, one of which was what was the maximum number of squares that could be coverd on a board with just one of each piece K,Q,R,B.N and P? These also included the old "proof" that 64 = 65 and multiple problems involving polyominoes

 Posts: 1640
 Joined: Thu May 15, 2008 1:37 am
 Contact:
Re: Knight's Tour: Are there other similar problems?
The knight's tour puzzle and two of those given by Paul B are also in The Complete Chess Addict (authors: Richard James  who posts on this forum  and Mike Fox).
Also in the book (and slightly different to Paul's problem 'e'):
"Place the eight white pieces on the board (keeping the bishops on oppositecoloured squares) so that:
i) the maximum number of squares are guarded;
ii) the minimum number of squares are guarded"
Also in the book (and slightly different to Paul's problem 'e'):
"Place the eight white pieces on the board (keeping the bishops on oppositecoloured squares) so that:
i) the maximum number of squares are guarded;
ii) the minimum number of squares are guarded"
Last edited by Angus French on Mon Feb 17, 2014 12:09 pm, edited 1 time in total.