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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## February 2023 - Challenge

In the eight queens problem, eight queens must be placed on a chessboard in such a way that no pair of queens is currently threatening one another. One possible solution to the eight queens problem is this:

In any solution to the eight queens problem, all the empty squares on the chessboard are threatened by at least two queens. We refer to a square that is threatened by exactly two queens as safe. These squares are marked here in green:

For the given configuration of queens, there is only one way to put four kings on the safe squares such that no two kings are currently threatening one another:

With apologies to standard chess notation, we can describe the locations of the queens and kings on this board using the following two lists of coordinates:
[(0, 2), (1, 5), (2, 3), (3, 1), (4, 7), (5, 4), (6, 6), (7, 0)]
[(1, 7), (3, 0), (7, 1), (7, 3)]

For n=14, an n\times n board and the following queen configuration:
[(0, 1), (1, 9), (2, 7), (3, 5), (4, 3), (5, 12), (6, 10), (7, 13), (8, 11), (9, 6), (10, 4), (11, 2), (12, 0), (13, 8)]

Is it possible to find arrangement of n kings. One such arrangement is
[(0, 0), (1, 10), (2, 0), (4, 0), (4, 2), (9, 2), (9, 9), (11, 0), (11, 7), (11, 11), (13, 5), (13, 9), (13, 11), (13, 13)]

There are 41 possible arrangements of n kings in total.

Your goal: Find a placement of n=20 queens on an n\times n board such that no pair of queens threatens each other, and the number of different ways to place n kings on the safe squares without any pair of kings currently threatening one another is 48.

Supply your answer in a list of the queen positions, as in the above example.

A bonus "*" will be given for finding a placement of n queens on an n\times n board such that no pair of queens is currently threatening one another, and there is no way to place n kings on the safe squares without any pair of kings threatening each other, for n\ge 26.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 02/02/2023 @ 10:00 AM EST
Solution: 05/03/2023 @ 12:00 PM EST
List Updated: 22/03/2023 @ 11:30 AM EST

*Lorenz Reichel(2/2/2023 3:06 PM IDT)
*Paul Revenant(2/2/2023 4:54 PM IDT)
*Amos Guler(3/2/2023 3:10 AM IDT)
*Bertram Felgenhauer(3/2/2023 6:03 AM IDT)
*Gary M. Gerken(3/2/2023 7:12 AM IDT)
*Alper Halbutogullari(3/2/2023 8:46 AM IDT)
*Reiner Martin(3/2/2023 1:59 PM IDT)
*Lorenzo Gianferrari Pini(3/2/2023 4:33 PM IDT)
Vamsi Krishna Talupula(3/2/2023 7:28 PM IDT)
Giorgos Kalogeropoulos(3/2/2023 8:29 PM IDT)
*Daniel Bitin(3/2/2023 11:34 PM IDT)
*Dieter Beckerle(4/2/2023 3:45 PM IDT)
*Daniel Chong Jyh Tar(4/2/2023 5:24 PM IDT)
Richard Gosiorovsky(4/2/2023 6:09 PM IDT)
*Todd Will(4/2/2023 6:47 PM IDT)
David Greer(4/2/2023 8:27 PM IDT)
V.Barbera(5/2/2023 1:08 AM IDT)
*Stefan Berger(5/2/2023 3:42 PM IDT)
*Hakan Summakoğlu(5/2/2023 7:59 PM IDT)
*Tim Walters(6/2/2023 2:09 AM IDT)
*David F.H. Dunkley(6/2/2023 2:42 AM IDT)
*Harald Bögeholz(6/2/2023 10:27 AM IDT)
Graham Hemsley(6/2/2023 11:22 AM IDT)
Dan Dima(6/2/2023 2:54 PM IDT)
*Guillaume Escamocher(6/2/2023 5:08 PM IDT)
*Latchezar Christov(7/2/2023 6:32 PM IDT)
*Bert Dobbelaere(7/2/2023 9:11 PM IDT)
*K S(8/2/2023 11:23 PM IDT)
*Martin Thorne(9/2/2023 7:30 PM IDT)
*Andreas Knüpfer(9/2/2023 9:19 PM IDT)
*Sanandan Swaminathan(11/2/2023 11:24 AM IDT)
Alex Fleischer(11/2/2023 3:01 PM IDT)
*John Tromp(12/2/2023 12:56 AM IDT)
Clive Tong(13/2/2023 10:32 AM IDT)
*Thomas Ilsche(13/2/2023 4:56 PM IDT)
*Stéphane Higueret(13/2/2023 10:55 PM IDT)
*Chris Shannon(14/2/2023 8:34 PM IDT)
*Phil Proudman(15/2/2023 5:10 PM IDT)
Sri Mallikarjun(15/2/2023 7:28 PM IDT)
*Mario Bielert(15/2/2023 9:38 PM IDT)
*Marco Bellocchi(16/2/2023 12:02 AM IDT)
Michael Liepelt(17/2/2023 8:20 AM IDT)
Ranjit Eswaran(18/2/2023 4:12 PM IDT)
*Phong Kah Ho(18/2/2023 9:09 PM IDT)
Prashant Wankhede(20/2/2023 3:36 AM IDT)
*Jean-Matthieu Schertzer(20/2/2023 12:47 PM IDT)
Jesús V(20/2/2023 8:25 PM IDT)
*Li Li(21/2/2023 7:28 PM IDT)
*Evert van Dijken(22/2/2023 12:58 PM IDT)
*Alexander Marschoun(23/2/2023 10:23 PM IDT)
*Nyles Heise(25/2/2023 6:12 AM IDT)
*Andreas Puccio(27/2/2023 12:42 PM IDT)
*Motty Porat(28/2/2023 7:33 PM IDT)
Jet Semrick(1/3/2023 12:53 AM IDT)
*Stephen Herbert Jr(1/3/2023 9:05 PM IDT)
John Goh(4/3/2023 5:15 PM IDT)
Virginia Ardévol Martínez(5/3/2023 11:14 PM IDT)
Elias Werner(13/3/2023 6:32 PM IDT)