Chapter 9, Problem 1E

To determine

The six positions for six nonattacking rooks on B and the corresponding SDR of A.

Answer to Problem 1E

The six positions for nonattacking rooks on B correspond to the SDR $\underset{\_}{\left(1,2,3,6,4,5\right)}$.

Explanation of Solution

Definition used:

Let Y be a finite set and $A=\left(A_1,A_2,\dots ,A_n\right)$ be a family of n subsets of Y. A family $\left(e_1,e_2,\dots ,e_n\right)$ of elements of Y is called a system of representatives of A, provided that $e_1$ is in $A_1$, $e_2$ is in $A_2$,…, $e_n$ is in $A_n$. If, in a system of representatives, the elements $e_1,e_2,\dots ,e_n$ are all different, then $\left(e_1,e_2,\dots ,e_n\right)$ is called a system of distinct representatives, abbreviated as SDR.

Calculation:

Given a chessboard B with forbidden positions.

Construct the rook family $A=\left(A_1,A_2,A_3,A_4,A_5,A_6\right)$ of subsets of $\left\{1,2,3,4,5,6\right\}$ on the board with the given forbidden positions as shown below in Figure 1.

From Figure 1, it is observed that there are a total of 16 forbidden positions out of 36 total positions.

In the above figure, each row has one of the labels $A_1,A_2,A_3,A_4,A_5,A_6$ and each column has one of the labels $1,2,3,4,5,6$.

These labels indicate that the family $A=\left(A_1,A_2,A_3,A_4,A_5,A_6\right)$ of subsets of $\left\{1,2,3,4,5,6\right\}$ is being associated, where $A_i$ is the set of columns in which the free squares in row i lies.

Thus, $A_1=\left\{1,2,3,6\right\}$, $A_2=\left\{2,3,5,6\right\}$, $A_3=\left\{2,3,4,5\right\}$, $A_4=\left\{6\right\}$, $A_5=\left\{4,5,6\right\}$ and $A_6=\left\{1,2,5,6\right\}$.

It is possible to place six nonattacking rooks on the given board if and only if the associated family A has an SDR.

One of the possible SDR for the given board is $\left(1,2,3,6,4,5\right)$.

The six nonattacking rooks that correspond to the SDR $\left(1,2,3,6,4,5\right)$ of $A^2$ is shown below in Figure 2.

From Figure 2, it is observed that the six nonattacking are possible the given family A has an SDR.