Problem $1.$ Let's say that we are using the queens algorithm to solve the $7-$Queens problem.

$($a$)$ If col $=\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $3$ & $6$ & $4$ & $7$ & $1$ & $5$ & $2\backslash \backslash$

$\backslash$hline & a

$\backslash$end$\{$tabular$\},$ then where is each queen located on the chessboard? Clearly indicate their locations in the picture below.

$($b$)$ If $\backslash (\backslash$mathrm$\{$col$\}=\backslash )\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $3$ & $6$ & $4$ & $7$ & $1$ & $5$ & $2\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\},$ what does promising$(5)$ return?

$($c$)$ If $\backslash (\backslash$mathrm$\{$col$\}=\backslash )\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $3$ & $6$ & $4$ & $7$ & $1$ & $5$ & $2\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\},$ what does promising$(6)$ return?

$($d$)$ If $\backslash (\backslash$mathrm$\{$col$\}=\backslash )\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $3$ & $6$ & $4$ & $7$ & $1$ & $5$ & $2\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\},$ what does promising$(7)$ return?

$($e$)$ Let's assume that the queens algorithm has just placed a queen in the fifth column of the first row $($see the col array below$).$ Write down what the col array looks like when it finds the next solution.

$\backslash [$

$\backslash$mathrm$\{$col$\}=\backslash$begin$\{$array$\}\{|$l$|$l$|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $5$ & & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$array$\}$

$\backslash ]$