The above picture shows a chessboard with two queens placed on it. As the queens do not share the same row, column or diagonal of the chessboard they are not attacking each other. Can you place another six queens on the board so that none of the eight queens are attacking each other? And if its possible, how many ways are there to do it?

This illustrated puzzle using a typical chessboard, an example of what is called the 8-queens completion problem, is from 1850. Yet only now, in a paper written by Chris Jefferson, Peter Nightingale and me published in the Journal of Artificial Intelligence Research, have we confirmed the depth of complexity hidden within the puzzle when scaled up to allow for boards of any size, with any number of queens pre-placed in any arrangement on the board a much harder version of the puzzle known as n-queens completion.

Unfortunately, due to misunderstandings when our paper was reported by the media (here for example, or here with correction) many people now think I am going to pay them US$1m. I am sorry to disappoint them, and hope here to put the record straight.

The n-queens completion puzzle is a form of mathematical problem common in computer science and described as NP-complete. These are interesting problems because if an efficient solution can be found for one NP-complete problem, it can be used to solve all NP-complete problems.

Some of these puzzles may seem unimportant identifying the largest number of Facebook friends that dont know each other, for example. But a fast and efficient solution to this problem could also be used to solve other problems with a more practical purpose for example, calculating the password used to encrypt data sent between a web browser and a bank. While it may seem odd that the placement of queens on a chessboard can in some way be translated to password encryption, that is indeed the case. That is the nature of all NP-complete problems.

Thousands of problems have been proved to be NP-complete. What I like about n-queens completion is that it is one of the simplest NP-complete problems to explain, especially to people who know the rules of chess. It is also a simple variant of one of the most widely studied problems in artificial intelligence: n-queens, which is the same puzzle but starting with an empty board rather than one with pre-placed queens. Following our paper, we now understand that the reason why the n-queens completion problem is so much harder than the version with an empty board is that it is an example of an NP-complete problem.

Nobody knows, even very roughly, how hard NP-complete problems are. They could be as easy as sorting a list of names into alphabetical order, or they could be exponentially harder. Finding out which they are is called the P vs NP problem, and it is one of the great unsolved mathematical problems so much so that the Clay Mathematics Institute (not me) is offering a prize of US$1m for the solution of P vs NP.

Since our paper shows that the n-queens completion problem is NP-complete, anyone able to show whether its an easy or difficult problem could win a million dollars. This seemed an obvious hook to publicise our paper, and while we were delighted to take part with Peter and I posing with giant chess pieces we only wish that the reporting hadnt given people the impression they could win the money for solving the n-queens problem, rather than the P vs NP problem that is far harder and potentially unsolveable.

Its possible the reason people misunderstood what was required to win the Clay Institutes prize is how many layers removed the prize requirements is from solving chess puzzles.

First, they needed to be tackling the right problem, since n-queens is easy and n-queens completion is hard.

Second, it is not enough to solve instances on a standard 8-by-8 chessboard. For example, we already know that the 8-queens completion problem from 1850 has two possible answers. People have to solve the problem for any sized chessboard.

The third layer is to solve the puzzle not just for a particular layout of queens, but for any possible layout of any possible number of queens on a board of any possible size. Even finding algorithms for this level of n-queens completion is not enough.

The fourth layer is to not just solve the puzzle, but to mathematically prove the properties of the algorithms that have given you the answer. This is where the prize money is: to solve the wider P vs NP question, one must either mathematically prove that an algorithm exists that can solve n-queens completion efficiently (technically, in polynomial time) or alternatively to mathematically prove that this is impossible. And, in either case, to have published this work in journals for the worlds mathematicians to pore over for two years.

Its possible that we hadnt made clear the sheer complexity of the task required to win the prize money. It might be said that we failed to explain these layers very well. If I may help those still aiming for the prize, however, I would advise the following:

Get a PhD in computational complexity

Be brilliant

Be very, very, lucky

Here is the original post:

Why the world’s toughest maths problems are much harder than a chess puzzle, and well worth US$1m – The Conversation UK

- Chess Puzzles 43-44 for Kids | Mate in 1 | Queen-Bishop ... - September 21st, 2017
- Can You Solve the Million-Dollar, Unsolvable Chess Problem? - Atlas Obscura - September 2nd, 2017
- Young Marape shines in Zimbabwe - Mmegi Online - August 30th, 2017
- Chess Ultra Makes A Move On Nintendo Switch - Nintendo Insider - August 7th, 2017
- Checkmate is possible when Chess Ultra comes to Switch - Vooks - August 6th, 2017
- Chess Ultra Coming To Nintendo Switch; New Trailer Released - DualShockers - August 2nd, 2017
- Small-Business Success Story: Learners Chess Academy - Kiplinger Personal Finance - August 1st, 2017
- Feature story: Lauren Goodkind plays, teaches and writes about chess - The Almanac Online - July 13th, 2017
- Solve Puzzles (And Sometimes Crime) In Picross Dating Sim - Kotaku - July 6th, 2017
- Play Chess For Your Soul In Chess Ultra - VRFocus - June 21st, 2017
- 8 must-see indie Xbox and PC games you may have missed at E3 2017 - Windows Central - June 20th, 2017
- Chess Ultra is now available for pre-order on Xbox One - OnMSFT (blog) - June 9th, 2017
- Chess Ultra release date announced for Xbox One to go alongside ... - TheXboxHub (blog) - June 7th, 2017
- Play the Grim Reaper and Save Your Soul in Chess Ultra This ... - VRFocus - June 7th, 2017
- On Chess: Are chess books dying? - St. Louis Public Radio - May 18th, 2017
- On Chess: Are Chess Books Dying? - KBIA - May 17th, 2017
- New data show most drivers use handheld devices - The Salinas Californian - April 28th, 2017
- Can You Outsmart Our Elementary School Math Problems? - FiveThirtyEight - April 7th, 2017
- Desktop v Mobile the best platforms for online gaming - Live in Limbo - March 23rd, 2017
- 'Chess Ultra' is Bringing its Luxurious Game Environments to PSVR, Oculus Rift, and HTC Vive - Road to VR - March 15th, 2017
- Shakira Ampaire all set for the Zonal Championships - Kawowo Sports - March 13th, 2017
- Chess Puzzles - 365Chess.com - November 25th, 2016
- Chess Problems, Puzzles, & Compositions - Chessopolis - November 15th, 2016
- Chess puzzle - Wikipedia - October 25th, 2016
- Chess Puzzles | Brilliant Math & Science Wiki - September 8th, 2016
- Home | Chess Puzzles! - June 20th, 2016
- Chess Puzzles | Solve Chess Problems and Compositions ... - September 28th, 2015
- IM Daniel Rensch Plays Beat The Clock With Chess Puzzles - Video - April 28th, 2015
- How to Solve Chess Puzzles #23 - Video - April 22nd, 2015
- How to Solve Chess Puzzles #22 - Video - April 17th, 2015
- Your Daily Chess Puzzles: April 8th, 9th and 10th 2015! - Video - April 13th, 2015
- Your Daily Chess Puzzles: April 5th and 6th 2015! - Video - April 10th, 2015
- Your Daily Chess Puzzles: April 3rd and 4th 2015! - Video - April 10th, 2015
- Solving Chess Puzzles #1 - Video - March 23rd, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #29 - Video - March 7th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #28 - "Forcing moves force cooperation!" - Video - February 24th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #26 - Video - February 19th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #27 - Video - February 19th, 2015
- How to Solve Chess Puzzles #21 - Video - January 31st, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #25 - Video - January 30th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #24 - Video - January 29th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #23 - Video - January 29th, 2015
- Alice: Madness Returns [Part 6] - Chess Puzzles - Video - January 18th, 2015
- The Best Chess Puzzles #33 The solutions of the chessbase.com Xmas puzzles - Video - January 11th, 2015
- How to solve Chess Puzzles: Chessworld.net Puzzle Practice #22 - Video - January 10th, 2015
- The best chess puzzles #32 The solutions of the chessbase.com Xmas puzzles #7-8 - Video - January 10th, 2015
- The best chess puzzles #31 The solutions of the chessbase.com Xmas puzzles #6 - Video - January 7th, 2015
- The Best Chess Puzzles #29 The solution of the chessbase.com Xmas puzzles #4 - Video - January 5th, 2015
- The Best Chess Puzzles #28 The solution of the chessbase.com Xmas puzzles #3 - Video - January 4th, 2015
- The best chess puzzles #30 The soluton of the chessbase.com Xmas puzzles. #5 - Video - January 4th, 2015
- 10 Daily Chess Puzzles #1 - Video - January 3rd, 2015
- The Best Chess Puzzles #27 The solution of the chessbase.com xmas puzzles #2. - Video - January 3rd, 2015
- 10 Daily Chess Puzzles #2 - Video - January 3rd, 2015
- The Best Chess Puzzles # 26 The solutions of the chessbase.com christmas puzzles #1 - Video - January 1st, 2015
- The Best Chess Puzzles #25 A cute mate in 3 for Xmas - Video - December 28th, 2014
- The best chess puzzles #24 a lovely mating attack. - Video - December 21st, 2014
- The best chess puzzles #22 A lovely kingside attack by Geller. - Video - December 16th, 2014
- The best chess puzzles #19 The hardest puzzle ever (?) - Video - December 3rd, 2014
- Tactical chess puzzles from 5 minute blitz games - Video - December 3rd, 2014
- Chess Puzzles #30 Checkmate in 3 moves - Video - November 27th, 2014
- The best chess puzzles #19 The "Benko-Riddle" - Video - November 26th, 2014
- The best chess puzzles #18 An easy mate in 4 by Lloyd. - Video - November 25th, 2014
- Cool Chess Puzzle #1 - Video - November 22nd, 2014
- The best chess puzzles# 17 A really fun study by the amazing Gady Costeff - Video - November 21st, 2014
- The best chess puzzles #16 Reti's amazing pawn ending revisited - Video - November 15th, 2014
- The best chess puzzles #14 A cute and clever mate in 3 puzzle for any skill level - Video - November 14th, 2014
- The best chess puzzles #14 A cute and easy mate in 4 puzzle by Anderson - Video - November 13th, 2014
- The best chess puzzles #13 The - 2 rooks 2 kings riddle- - Video - November 11th, 2014
- The best chess puzzles #12 A lovely study by Pogosiatns - Video - November 8th, 2014
- How to Solve Chess Puzzles #20 - Video - November 7th, 2014
- The best chess puzzles #11 The fastest possible stalemate from the starting position + a surprise:) - Video - November 7th, 2014
- The best chess puzzles #10 A fantastic mate in 3 by Grimshaw - Video - November 5th, 2014
- The best chess puzzles #9 Another lovely study with an unusal draw - Video - November 2nd, 2014
- The best chess puzzles #8 A brilliant study by Hurtig - Video - November 1st, 2014
- How to Solve Chess Puzzles #19 - Video - October 31st, 2014
- The best chess puzzles #7 Construct a game where black delivers mate on move 5 by playing Rxh1 - Video - October 30th, 2014
- The best chess puzzles#6 - Video - October 28th, 2014
- The best chess puzzles #5 A mate in 5 puzzle composed by Firtz Giegold - Video - October 27th, 2014
- chess puzzles onlne 3 - Video - October 27th, 2014
- Chess puzzle - Wikipedia, the free encyclopedia - October 27th, 2014