The game

Given an   N×N   square array of zeroes or ones in an initial configuration,   and a target configuration of zeroes and ones.



The game is to transform one to the other in as few moves as possible by inverting whole numbered rows or whole lettered
columns at once   (as one move).

In an inversion.   any  1  becomes  0,   and any  0  becomes  1  for that whole row or column.



Task

Create a program to score for the Flipping bits game.

 The game should create an original random target configuration and a starting configuration.
 Ensure that the starting position is never the target position.
 The target position must be guaranteed as reachable from the starting position.   (One possible way to do this is to generate the start position by legal flips from a random target position.   The flips will always be reversible back to the target from the given start position).
 The number of moves taken so far should be shown.


Show an example of a short game here, on this page, for a   3×3   array of bits.




sub MAIN ($square = 4) {
    say "{$square}? Seriously?" and exit if $square < 1 or $square > 26;
    my %bits = map { $_ => %( map { $_ => 0 }, ('A' .. *)[^ $square] ) },
        (1 .. *)[^ $square];
    scramble %bits;
    my $target = build %bits;
    scramble %bits until build(%bits) ne $target;
    display($target, %bits);
    my $turns = 0;
    while my $flip = prompt "Turn {++$turns}: Flip which row / column? " {
        flip $flip.match(/\w/).uc, %bits;
        if display($target, %bits) {
            say "Hurray! You solved it in $turns turns.";
            last;
        }
    }
}
 
sub display($goal, %hash) {
    shell('clear');
    say "Goal\n$goal\nYou";
    my $this = build %hash;
    say $this;
    return ($goal eq $this);
}
 
sub flip ($a, %hash) {
    given $a {
        when any(keys %hash) {
            %hash{$a}{$_} = %hash{$a}{$_} +^ 1 for %hash{$a}.keys
        };
        when any(keys %hash{'1'}) {
            %hash{$_}{$a} = %hash{$_}{$a} +^ 1 for %hash.keys
        };
    }
}
 
sub build (%hash) {
    my $string = '   ';
    $string ~= sprintf "%2s ", $_ for sort keys %hash{'1'};
    $string ~= "\n";
    for %hash.keys.sort: +* -> $key {
        $string ~= sprintf "%2s ", $key;
        $string ~= sprintf "%2s ", %hash{$key}{$_} for sort keys %hash{$key};
        $string ~=  "\n";
    };
    $string
}
 
sub scramble(%hash) {
    my @keys = keys %hash;
    @keys.push: | keys %hash{'1'};
    flip $_,  %hash for @keys.pick( @keys/2 );
}