For the formulation of a modified N queens.
Unlike the original Queens problem, there is just one rule-all N queens must be placed row-wise first.
The goal is to select the smallest integer $p$ such that $p^2 \geq N$.
I am thinking this boolean expression for x_i_j, which represents $i$-th row and $j$-th column:
x_1_1 & x_1_2 & ...x_1_p &
x_2_1 & x_2_2 & ...x_2_p &
... &
... &
x_p_1 & x_p_2 & ...x_p_p
Is the expression correctly represented? Or there will be 'or' instead of '&'. Any clue/help will be appreciated.
Your expression does not seem to be correct, the way you wrote it will evaluate to boolean true only if there is queen on every position on the $p \times p$ board. What you want instead is to identify a location of last queen so that $N$ queens are placed, say it will be on position $i,j$ (we must have $(i-1)p+j=N$). Then we want all $x_{a,b}$ with $a \leq i$ or $a=i$ and $b \leq j$ to evaluate to true. The rest of the positions will have no queen, so $x_{a,b}$ should evaluate to false there. If you write it in a little matrix, you want
\begin{array}{c|cc} &1&2&\dots&j&j+1&\dots&p \\ \hline 1&true&true&\dots&true&true&\dots&true\\ 2&true&true&\dots&true&true&\dots&true\\ \vdots\\ i-1&true&true&\dots&true&true&\dots&true\\ i&true&true&\dots&true&false&\dots&false\\ i+1&false&false&\dots&false&false&\dots&false\\ \vdots\\ p&false&false&\dots&false&false&\dots&false \end{array}
So now just connect these with logical and, use logical negation where the variable needs to evaluate to false, and you should get something like this:
$$ x_{1,1} \land x_{1,2} \land \dots \land x_{1,p}\\ \land x_{2,1} \land x_{2,2} \land \dots \land x_{2,p}\\ \vdots\\ \land x_{i,1} \land x_{i,2} \land \dots \land x_{i,j} \land \lnot x_{i,j+1} \land \dots \land \lnot x_{i,p}\\ \vdots\\ \land \lnot x_{p,1} \land \lnot x_{p,2} \land \dots \land \lnot x_{p,p}. $$