| MANSW | The Mathematical Association of New South Wales, Inc. Promoting Quality Mathematics Education for all. |
MATHSEARCH99 Project99
Prepared by Geoff Ball and Humphrey Gastineau-Hills, University of Sydney.
Dongboard Counting - page 5
You may by now be aware that different shaped donghoards can have
the same rook polynomial.
Definition 4. Equivalence. Two dongboards, A and B, that have
the same rook polynomial are said to be equivalent.
If dongboards A and B are equivalent we write A ~ B.
Exercise 10. Determine if the donghoards in Figure 9
are equivalent.
|
Figure 9 |
|
Exercise 11.
(a) Construct a 4-row dongboard C1 with 8 cells in which no two rows are identical.
Note: Two rows may have the same number of cells, but the existence and placement of spaces may make them different.
(b) Find three dongboards that are equivalent to the dongboard C1 of part (a) of this question.
There are a number of different directions we could go from here
so you have the choice of either a more geometrical or algebraic
research effort.
Alternative A. Further investigation of equivalent
dongboards.
We have seen that equivalent dongboards can have quite different
shapes.
However, it is worth considering if there are any underlying
geometric connections.
In particular you should investigate the effect of geometric
transformations on the rook polynomial of a dongboard.
In your investigation you should at least consider whether such
transformations as:
- row and/or column permutations, i.e., rearranging the rows and/or columns;
- transposition, i.e., rows become columns and columns become rows;
- geometric transformations such as reflections and rotations.
result in different rook polynomials.
