This blog is going to take the place of our third blog post.
The one topic that I haven’t covered yet is communicating mathematics. I doubt
this will be as controversial of a piece as the previous post was since it will
be more informative and less opinion based. So if one is reading this looking for
something similar, I apologize.

In looking back through our course page on different
thinkers around the time of the third blog post I was drawn to Thabit ibn Qurra
since I knew so little about him. I wanted to delve into his mathematics for my
blog. Looking at his Wikipedia page, I noticed a link to a chess solution. From
that page I found a link to the eight queens puzzle. Following my interest and Wikipedia
links I decided to settle on this as my topic for this blog.

The problem is as follows. Is there a way to place eight
queens on a standard chessboard so that none of them could attack each other
either diagonally, horizontally, or vertically? The history of this problem is
part of my interest.

Max Bezzel first introduced the puzzle in 1848 in a German
magazine. Over the next six years there were a variety of solutions published
in various magazines. With a reposing of the question the problem shifted to a
question of how many different solutions to the problem there were. Franz Nauck
was the one responsible for this reposing and asserts that there were 60 solutions
to the problem. This sparked a debate between two friends Gauss and Schumacher.
They communicated with letters for around a month. During that time Nauck
corrected himself and affirmed that there were actually 92 solutions, but
offered no proof of this assertion. Gauss referred to Nauck’s correction and
laid out how to go about proving it, though never actually proves it. It seems
that this task wasn’t of extreme importance to Gauss since he didn’t spend the
time to solve it. These letters were published about fifteen years later.

The
solution that Gauss proposed was clever. I’ll try to communicate as best I can
the solution that was proposed. Gauss reformulates this problem into an arithmetic
one. Gauss begins by assigning a position to each permutation of the numbers
one through eight. Thus 12345678 would have the first queen at a1, the second
queen at b2, the third queen at c3, and so on. The permutation 45632178 would
have the first queen at a4, the second queen at b5, the third queen at c6, and
so on. Using this idea of position Gauss was able to work with the solutions
more easily. He found that if each permutation was summed with 12345678 and separately
with 87654321, and if these sums were all different in each number that this
was a solution to the problem. This is difficult to understand without an
example so I will share the example that Gauss sent Schumacher.

All
these sums are different in this permutation of 15863724. This means that it is
a solution to the eight queens problem. In the next permutation of 13425867 we
can see that 10, 9, and 8 all appear twice. This means that it is not a
solution to the problem.

By setting the problem up this way Gauss was able to more
quickly sift through solutions. The setup itself prevents queens from lying on
the same row or column. The addition of the permutation 12345678 prevents
queens from lining up diagonally to the right; the addition of the permutation
87654321 prevents the queens lining up diagonally to the left.

The solution isn’t flushed out arithmetically that I could
find, but was explained to have 12 fundamentally different solutions that,
through rotations, are flushed into all 92 solutions. This geometric solution
is nice, but I wanted the arithmetic solution.

The problem continued to evolve into boards of differing
sizes and got into complications that I won’t flush out. As stated earlier,
Gauss’ passion didn’t seem to be evident in this problem since he didn’t bother
to create every permutation, or offer an argument for why there were only 92
solutions. I wish that his argument had been clearer and that Gauss had had
more passion in the subject. This was a fun problem to post on because it
combines my passion for chess and math!

I have a good chess math teacher story if you're interested.

ReplyDeleteNice exposition. Did you try the problem before reading up on solutions? (I never have. GeoGebra if I had time....) That would be an interesting addition. An image of one of the 12 would be interesting here. (complete)

Other Cs: +

Flushed?