# The N + k Queens Problem Page

## Background

In 1848, Max Bezzel first posed the Eight Queens Problem: Place eight queens on a chessboard so that no two queens attack each other. This problem was generalized in 1869 to the N-Queens problem (N mutually non-attacking queens on an N-by-N chessboard).

In 1995, Michael Anshel of the City University of New York presented the following problem to an Artificial Intelligence seminar : if we remove some of the squares of the chessboard, can we place more than N queens on the "partial chessboard" that remains? In 1998, Anshel's student, Kaiyan Zhao, defended her doctoral thesis entitled "The Combinatorics of Chessboards," which considered "The Queens Problem on a Partial Chessboard". Among other things, Zhao proved that three pawns are necessary and sufficient to allow six nonattacking queens on a five-by-five board.

In 2004, the Chess Variant Pages sponsored a contest that posed the following question (credited to Wim Henk Boedlander): We can put nine mutually non-attacking queens on an eight-by-eight board if we put some pawns on the board to block queen attacks. How few pawns are necessary? The answer turns out to be 1, as you can see in the picture below. The Nine Queens Problem contest can be found at http://www.chessvariants.org/problems.dir/9queens.html .

The contest participants soon asked about generalizations, such as "How many pawns would it take on a larger board?" It turns out that for N > 5, we need to put only one pawn on the board in order to allow us to place N+1 nonattacking queens on the board.   More generally, for each k > 0, if N is large enough we can place N + k queens and k pawns on an N-by-N board so that none of the queens attack each other. For more details, see our papers listed below .

### Open Problems

Here is a partial list of unsolved problems related to the N+k Queens Problem:

• For a particular N and k, how many ways are there to place N + k queens and k pawns on an N-by-N board so that no two queens attack each other? We've done some computer searches and come up with some numbers, but we haven't got a closed-form formula. (Then again, no formula has been found yet for the number of solutions to the N-Queens problem.)
• Where can the pawns go?
• What happens if we look at non-square boards?
• On a standard chessboard, if you want to place queens on the board so that every square is attacked or occupied, the minimum number of queens you need is five. If we wanted to increase or decrease that number, how many pawns would we need to place on the board, and where? (And how many pawns would we need on other-sized boards to increase or decrease the number of queens needed to dominate those boards?)
• What happens to other graph-theoretic parameters when we place pawns on a chessboard?
• What happens if we use pieces other than queens, such as bishops or rooks or even nonstandard pieces like the amazon (combination of queen and knight)?

Some partial answers are available in the papers placed on this site.

### Talks

Here are slides from some of our talks related to this subject:

• SeparationNumbersEKU.pdf : I gave a colloquium talk to the Department of Mathematics and Statistics at Eastern Kentucky University on September 29, 2006.
• tenn_pres.ppt and finalcap.ppt : Matthew Wolff's talks, the first was given at the 19th Annual Cumberland Conference and the second was his senior capstone presentation. These presentations discuss how we used Knuth's Dancing Links algorithm to develop a faster parallel program for counting the number of solutions to the N+k Queens Problem. (Note: The files are in MS-PowerPoint format.)
• WahleSUMS.ppt: On October 13, 2007, Nick Wahle gave a talk at the Shenandoah Undergraduate Mathematics and Statistics Conference on transit graphs, a generalization of chessboard graphs. This file is in MS-PowerPoint format.
• HuffordSUMS.ppt: On October 13, 2007, Casey Hufford gave a talk at the SUMS conference on book embeddings of chessboard graphs with some results on the effect of placing a pawn on the board.. This file is in MS-PowerPoint format.
• MillerMCCC.ppt: On October 13, 2007 (apparently a busy day for us), John Miller gave a talk at The Twenty-first Midwest Conference on Combinatorics, Cryptography and Computing comparing the performance of algorithms that count the number of solutions to the N+k Queens problem. This file is in MS-PowerPoint format.
• Reflections.ppt: I gave a talk on August 1, 2008 at the MAA MathFest in Madison, Wisconsin on symmetric solutions to the N+k Queens problem. This file is in MS-Powerpoint format.
• chatham_centro_queens.pdf: On August 6, 2011, I gave a talk at the MAA MathFest in Lexington, Kentucky reporting further progress on solutions with various sorts of symmetry. Here's a handout: mathfest2011rdchandout.pdf.
• KYMAA_2013_final_v2.pptx: For the past few years, our Undergraduate Research Fellows have been looking for nicer solution patterns. This presentation was given at the 2013 Annual Meeting of the Kentucky Section of the Mathematical Association of America and at the 2013 University of Tennessee Undergraduate Conference in Knoxville.

### Papers

Here are preprints of papers we've written:

• queenssep.pdf : "The Queens Separation Problem" published in Volume 69 of Utilitas Mathematica in March 2006.
Note: Table 2 in this paper is faulty. See the next paper or go here for correct numbers.
• QueensSep2.pdf: "Independence and Domination Separation on Chessboard Graphs", published in Volume 68 (2009) of the Journal of Combinatorial Mathematics and Combinatorial Computing.
• cmj204-210.pdf: "Reflections on the N + k Queens Problem", published in the May 2009 issue of The College Mathematics Journal.
• dlxMCCC.pdf: "Algorithm Performance for Chessboard Separation Problems", published in Volume 70 (2009) of the Journal of Combinatorial Mathematics and Combinatorial Computing.
• JDLXFinalCopy.doc: " JDLX: Visualization of Dancing Links", a paper on a Java applet demonstrating the DLX algorithm presented on September 26, 2008 at the 2008 Midwest Regional Conference of the Consortium for Computing Sciences in Colleges.
• centrosymm2.pdf: "Centrosymmetric Solutions to Chessboard Separation Problems", published in the Bulletin of the Institute of Combinatorics and its Applications, volume 65 (2012).

### Example Solutions

If you want to see some solutions to the N+k Queens Problem, look at the following files:

• nopawn.pdf: A PDF file with pictures of the fundamental solutions to the N+0 Queens Problem :) for N=4,5,6,7,8,9, and 10.
• onepawn.pdf: A PDF file with pictures of the fundamental solutions to the N+1 Queens Problem for N=6,7,8,9, and 10.
• twopawn.pdf: A PDF file with pictures of the fundamental solutions to the N+2 Queens Problem for N=7,8,9, and 10.
• threepawn.pdf: A PDF file with pictures of the fundamental solutions to the N+3 Queens Problem for N=8,9, and 10.
• fourpawn.pdf: A PDF file with pictures of the fundamental solutions to the N+4 Queens Problem for N=9,10, and 11.
• twopawngap.txt: A text file that gives some solutions to the N+2 Queens Problem for N between 11 and 100.
• threepawngap.txt: A text file that gives some solutions to the N+3 Queens Problem for N between 8 and 100.

### Programs

Here are some recreational scripts involving the N+k Queens Problem:

Here is an educational Java applet by Maureen Doyle, Amber Rogers, and Bernadina Rawe illustrating the DLX algorithm as applied to the N-Queens Problem: JDLX.jar. (We used a DLX algorithm to count the number of solutions to the N+k Queens Problem.)

### Solution Counts

If you want the number of solutions to the N+k Queens Problem for certain small N and k, try one of these links:

### Outside Pages

Here are outside websites on matters related to the N + k Queens Problem: