# n-Queens Completion is NP-Complete

### Update, 2021

Over the years since we published this research, many people have approached us having solved the n queens puzzle, either for one n like 8 or 1000, or having written an algorithm to solve it for different sizes.  Unfortunately this is not a major result in Computer Science and does not make one eligible to claim the \$1M Clay prize. Many have been disappointed by this so we want to clarify why  this is the case.

It is true that work on this problem could potentially result in the award but only if some exceptionally difficult conditions are met.

• EITHER prove mathematically that NO possible algorithm could solve the n queens completion problem in polynomial time;
• OR prove that there is an algorithm which is guaranteed to solve every instance of the n queens completion problem in polynomial time. Note that in this case the algorithm has to work on the completion version of the problem studied in our paper, not placing queens on an empty board; the algorithm has to give the correct answer on every possible instance given to it; and there has to be a mathematical proof that the algorithm’s runtime is bounded above by some polynomial in the size of the board.  However fast a given algorithm runs when tested, this is not sufficient because there are an infinite number of possible tests available, so a mathematical proof is required.
• AND in either case, prove this at a level that is published in a respected academic source and is widely accepted by research experts as correct.

We are delighted that our work has led so many people to be interested in the problem of solving problems like the n queens puzzle that fascinates us.  But we also apologise for any impression we gave, unintentionally, that a solution to the n queens puzzle could lead to the award of the prize except under the extremely strenuous conditions listed above.

Ian Gent, 10 May 2021

Original Post from 2017:

Ian Gent, Christopher Jefferson and Peter Nightingale have shown that a classic chess puzzle is NP-Complete. Their paper “Complexity of n-Queens Completion” was published in the Journal of Artificial Intelligence Research on August 30, 2017.

The n-Queens puzzle is a classic chess problem: given a chessboard of size n by n, can you place n queens so that no two queens attack each other?  That is, can you place the queens with no two queens are on the same row, column, or diagonal? The n-Queens puzzle has long been known to be simple to solve:  you can solve the problem for all n except 2 and 3, and solutions for all other n can be described in a few lines.  This very simplicity has led to repeated

Peter Nightingale and Ian Gent at Falkland Palace, Wednesday, 17 August 2017.