April 2024 Issue 5
Newsletter
Editors: IM Jason Wang & FM Erick Zhao
Chess for Unity
Eight Queen Puzzles
By Jason Wang
A Report on the Free Workshop
By Erick Zhao
Eight Queen Puzzles
By IM Jason Wang
In the December edition of our newsletter we analyzed the Knight’s Tour, a puzzle that has captivated the scientifically-minded chess community involving a knight visiting each of the sixty-four squares on a chessboard. In this edition, we introduce a puzzle of equal analytical merit: the Eight Queens Puzzle. The puzzle, originally proposed by composer Max Bezzel in 1848, asks the solver to place eight queens on a chessboard such that no two queens are threatening each other.
Since then, analysis of this puzzle has boomed to larger and larger boards, algorithms for finding solutions, and various variants of this problem. Like the Knight’s Tour, it has been studied in both computer science and mathematics.
Let’s start with a simpler example: placing eight rooks on the chessboard such that no two rooks attack each other. How many ways are there to do this? This is a problem in mathematical combinatorics, and we invite the reader to give this problem a try.
Let’s reason through this. Since rooks move horizontally and vertically, clearly no two rooks can lie on the same row or column. Therefore, we must have exactly one rook on every row, and exactly one row on every column. If we start on the a-file, there are 8 possible squares to place a rook; any of them will work. Then, for the b-file, there are 7 possible squares to place the square — any square that is not on the same row as the first rook. Continuing this, we see that there are 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1=40,320 ways to place these eight rooks, which mathematicians write as 8!, where ‘!’ represents the factorial symbol.
Returning to the Eight Queens Puzzle, there are clearly at most 40,320 ways to place the eight queens such that no two attack each other, since the queen is able to capture diagonally as well. This combinatorial argument results in a simplistic but relatively decent reduction in computational power required: it is much faster for the computer to analyze these 40,320 cases than to brute force through the 4 billion plus arrangements of eight queens on a sixty-four square chessboard.
With this in mind, it has been shown that out of these 40,320 possibilities, there are 92 possible solutions to the Eight Queens Puzzle. Out of these 92, twelve are unique up to rotation and reflection, meaning that none of these twelve can be transformed into another of these twelve with simple rotations and reflections of the chessboard.
How about boards of other sizes? The mathematician, of course, is not content with analyzing only the 8×8 board. The puzzle generalizes to an n x n board easily: place n queens on the board such that no two queens are attacking each other. First, let’s try it with some smaller, more manageable cases!
Clearly for a 1 x 1 board there is 1 solution; for 2 x 2 and 3 x 3 boards there are no solutions. For a 4 x 4 there are 2, for a 5 x 5 there are 10, and so on.
Is there a pattern for larger and larger n? Sadly no, and the highest value of n for which the number of solutions has been computed is n=27, with over 29 quadrillion solutions. Although exact values for larger values of n have not yet been computed, the number of solutions can be approximated as (0.143n)n; our definitions of “large” and “approximately” are loose, but we will leave it to the interested reader to delve deeper if they wish.
Whew! That was quite an introduction. But again, it is only an introduction to this fascinating puzzle, and there is so much more to explore. Algorithms for finding solutions, generalizations to different topological boards … the ocean of exploration for the Eight Queens Puzzle is bountiful.
Lastly, I’d like to address one common question regarding puzzles like the Knight’s Tour and the Eight Queens Puzzle, which is why? What exactly is the point of analyzing these seemingly meaningless results?
The answer is two-fold. Firstly, the simplicity of the Knight’s Tour and the Eight Queens Puzzle combined with the possible insights gleaned from them make them something worth grasping the basics of. But of course, the most important reason is that chess is meant to be fun, and the brilliance and versatility of pieces like the knight and queen should be appreciated at heart.
The 1st Ever Special Chess for Unity Workshop
By FM Erick Zhao
To many high school students, it seemed like just another Friday afternoon—a respite from the usual burden of homework, tests, and impromptu tasks. However, for the young chess players in elementary school and middle school, there was no pause in their focus. The 2024 National Elementary Championship and National Middle School Championship, the pinnacle of the year’s tournaments, were on the brink of beginning.
As board members of Chess for Unity, we recognized this moment as an opportunity to support and prepare these aspiring players. Thus, we decided to use our resources and community to host our first ever Chess for Unity special workshop.
Over 50 elementary school students from across the United States and Canada gathered on the innovative Zoom platform to take part in this complimentary event. Board members FM Arthur Xu, FM Vincent Tsay, IM Liran Zhou and FM Erick Zhao led the three available sections.
Instead of the usual lecture format, we opted to transform this workshop into an interactive and enjoyable experience. We organized simultaneous exhibitions (where Erick was completely destroyed), puzzle battle competitions, and a variety of engaging tactical exercises!
Overall, it was an incredible experience both for the teachers and the students, and we hope to do more of these workshops in the future!
(IM Jason Wang took part in our workshop planning meetings though he was not able to attend it due to a schedule conflict.)
Two tactics that Liran and Erick used at the workshop.
View more tactics from them: puzzles (20 chapters); puzzles (9 chapters).