https://github.com/onecricketeer/rookpoly
RookPolynomial solver
https://github.com/onecricketeer/rookpoly
Last synced: 3 months ago
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RookPolynomial solver
- Host: GitHub
- URL: https://github.com/onecricketeer/rookpoly
- Owner: OneCricketeer
- Created: 2012-07-13T19:04:47.000Z (almost 13 years ago)
- Default Branch: master
- Last Pushed: 2016-11-07T21:23:24.000Z (over 8 years ago)
- Last Synced: 2025-01-21T09:11:35.504Z (4 months ago)
- Language: C#
- Size: 22.5 KB
- Stars: 1
- Watchers: 1
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
RookPoly
========
#### Watch it on Youtube! (click the image)
[](http://www.youtube.com/watch?v=5HjWGSzAq4Q)
A Rook Polynomial mathematically describes the number of ways to place
0 ... (the lesser of m and n) rook chess pieces on a m x n chess board.
The idea is that it is easier to calculate the total number of rooks that
can be placed instead of counting each number individually.
Ex. 1 + 81x + 9x^2 says there is 1 way to place 0 rooks, 9 ways to place 2
and 81 ways to place 2 rooks. From this knowledge, you can infer the board
is a 9 x 9 square.
My solution to this problem was a good lesson in recursion, as that is how
we solved these problems in one of my math classes.
For example,
We start with 1 way to place 0 rooks on any board.
Then we acquire more cases, which will now on be described in polynomial notation as above.
```
+---+
| | = 1 + x
+---+
"1 + x" mathematically represents 1 way to place 0 rooks, and 1 way to place a single rook.
+---+---+
| | | = 1 + 2x
+---+---+
+---+---+
| | |
+---+---+ = 1 + 3x + x^2
| |
+---+
+---+---+
| | |
+---+---+ = 1 + 4x + 2x^2
| | |
+---+---+
+---+
| | +---+
+---+---+ = 2 * | | = 2*(1+x)
| | +---+
+---+
```
... and so on.
Using these we can start with any board and form a polynomial representing it.
In order to do so, we must select a single tile, then delete it from the board, multiply the new board's
polynomial by x, and add that result to a new board formed by deleting the row and column of the selected
cell.
For example
```
+---+---+---+
| | | |
+---+---+---+
| | |
+---+---+
+---+---+---+ +---+---+---+
| X | X | X | | | | X |
+---+---+---+ + x * +---+---+---+
| | | | | |
+---+---+ +---+---+
+---+---+
+---+---+ | | |
| | | + x * +---+---+
+---+---+ | | |
+---+---+
```
Using our base cases, we now get 1 + 2x + x*(1 + 4x + 2x^2) = 1 + 3x + 4x^2 + 2x^3.
The process of determining the polynomial within the program breaks the board down into a single tile,
causing the calculation to take slower than it would if I had determined a way for it to recogize base
case shapes. In an effort to increase the speed, I threaded the calculation process.
Further work on this project would include a way to view the possible ways to place these combinations of rooks.