This can be extended to create a random solution, just randomize the initial block. (This is similar to Abigail-II's solution). Here @start is a N by N matrix whose entries are a random permutation of {1, .., N^2}.

my $n     = shift || 3;

my @start = do {
    use List::Util 'shuffle';
    my @perm = shuffle 1 .. $n*$n;

    map {
         my $row = $_;
         [ map { $perm[$row * $n + $_] } 0 .. $n-1 ]
    } 0 .. $n-1;
};

my @solution;

for my $x ( 0 .. $n*$n - 1 ) {
    for my $y ( 0 .. $n*$n - 1 ) {

        my $block_x = int ($x / $n);
        my $block_y = int ($y / $n);

        $solution[$x][$y] =
            $start[ ($block_x + $y) % $n ][ ($block_y + $x) % $n ];
    }
}

print join "" => map { "@$_\n" } @solution;
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