The problem
was originally published by Max Bezzel in the mid eighteen hundreds. Bezzel was
a both a mathematician, and a chess enthusiast. It took two years for solutions
to appear. Franz Nauk released them. Nauk then went to continue the problem
into a more generalized state of understanding. The problem was modified from
having eight queens to having "N" number of queens. After changing
the number of queens Nauk modified the dimensions of the board. Instead of the
traditional eight by eight board, it became "N" by "N"
boards. "N" is not only the length, and width, but also the number of
queens on the board. Mathematicians such as Gauss and Gunther have since
contributed to the problem.
Looking
primarily at the standard value of eight for "N", this meaning the
board is a standard eight by eight chessboard, and there are eight queens on
the board. The problem asks, if possible can all "N" number of queens
be on the board and not be threatened by another queen? Color of the queen does
not matter in this case, and if it can be done, how many possible ways can it
be done? A queen in chess is considered to be the single most powerful piece,
as it is the most versatile piece to move around the board. It can move both
horizontally, vertically, and across diagonals. This makes positioning eight of
them without threatening another difficult on a confined space. There are
solutions to the standard eight by eight board. Starting with the total possible
ways that eight queens can be positioned on a eight by eight board is the first
part. There are sixty-four tiles on a standard chessboard. That means there are
sixty-four factorial possible tile positions. If eight of the positions are
taken up with queens that leaves fifty-six factorial tile positions. Since the
queens are interchangeable among their given positions eight factorial also
comes into play. So to find the number of ways that eight queens can be
positioned on a chessboard, the total, sixty-four factorial, would be divided
by 56 factorial multiplied by eight factorial. A factorial refers to that
number multiplied by every previous whole number before that number. For
example six factorial would be six, times five, times four, times three, times two, times
one. A factorial has the nice property of showing how many ways you can arrange a
number of times. For example if you wanted to know how many ways to arrange
three items, the answer would be three factorial, or six. That makes numbers
such as sixty-four factorial an immensely large number. Same with respect to fifty-six
factorial. The net calculation on the number of ways to arrange the eight
queens is roughly around four billion. This includes all permutations, but
without the restriction of non-intersection between queens. Factoring that in
reduces the number down to 92, continuing the sifting of possible outcomes, by
removing redundancies due to reflections and rotations, the number goes down to
twelve. There are twelve possible solutions to the eight-queen problem.
To put factorials into scale of
how big these numbers are, fifty-two factorial is less then the number of
possible tiles, sixty-four factorial, and the number on non queen tiles fifty
six factorial. Fifty-two factorial is the number of possible ways a deck of
cards can be arranged. This is chosen because there is rather nice statistic
about fifty-two factorial, and it is smaller then some of the numbers used in
this problem. To understand fifty-two factorial, first stand on the equator.
Start playing solitaire with a deck of cards, assume every new game is a new
deck, and you win every game. Every billion years take a step forward, which
should allow for a couple trillion games every step. Every time you walk around
the earth remove a drop from the Pacific Ocean. Once the ocean is empty start
stacking pieces of paper on the bottom. Once you have reached the sun, roughly one
hundred million miles away. Notice not all decks have been played yet, so
remove the papers one at a time, and put the water back in the ocean drop by
drop, and go to and from the sun a thousand more times. Now about a third of 52
factorial decks have been played. That is how big fifty-two factorial is.
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