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##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$}} ##}}$nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn$}} ##}}$n++++++++++++++++++++++++++++++++++n$}} ##}}$n+================================+n$}} ##}}$n+=$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$=+n$}} ##}}$n+=$kkkkkkkkkkkkkkkkkkkkkkkkkkkk$=+n$}} ##}}$n+=$k==========================k$=+n$}} ##}}$n+=$k==========================k$=+n$}} ##}}$n+=$k==4444444444444444444444==k$=+n$}} ##}}$n+=$k==4********************4==k$=+n$}} ##}}$n+=$k==4*$$$$$$$$$$$$$$$$$$*4==k$=+n$}} ##}}$n+=$k==4*$ssssssssssssssss$*4==k$=+n$}} ##}}$n+=$k==4*$s;;;;;;;;;;;;;;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<<<<<<<<<<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<<<<<<<<<<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<''''''''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<''''''''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<'';;;;''<<;s$*4==k$=+n$}} }}$n+=$k==4*$s;<<'';##;''<<;s$*4==k$=+n$}} }}$n+=$k==4*$s;<<'';##;''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<'';;;;''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<''''''''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<''''''''<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<<<<<<<<<<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;<<<<<<<<<<<<;s$*4==k$=+n$}} ##}}$n+=$k==4*$s;;;;;;;;;;;;;;s$*4==k$=+n$}} ##}}$n+=$k==4*$ssssssssssssssss$*4==k$=+n$}} ##}}$n+=$k==4*$$$$$$$$$$$$$$$$$$*4==k$=+n$}} ##}}$n+=$k==4********************4==k$=+n$}} ##}}$n+=$k==4444444444444444444444==k$=+n$}} ##}}$n+=$k==========================k$=+n$}} ##}}$n+=$k==========================k$=+n$}} ##}}$n+=$kkkkkkkkkkkkkkkkkkkkkkkkkkkk$=+n$}} ##}}$n+=$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$=+n$}} ##}}$n+================================+n$}} ##}}$n++++++++++++++++++++++++++++++++++n$}} ##}}$nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn$}} ##}}$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} 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##}}}prrrrrrrrrrrrrrrrrrrrrrrrrrp}}} ##}}}pppppppppppppppppppppppppppp}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} ##}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} 

As you can see, every single character is part of a square. The top half of each big square is commented out by line comments, and the bottom half (including the second non-commented line) is escaped by a big here-document beginning in the middle of the square (<<'';#) and ending at the empty line after the big square.

This program was automatically generated from the following program:

($r,$c,@a)=@ARGV; for$y(0..$r-1){ for$x(0..$c-1){ for$s(1..$c-$x-1){ $m=$a[$o=$c*$y+$x];$k=0; for$t(1..$s){ for($t,$c*$t,$t+$c*$s,$c*$t+$s){ $k+=$a[$o+$_]eq$m; }}$n+=$k==4*$s; }}}print$n; 

Which is a program to solve the same challenge, but (very lightly) golfed for bytes and short line lengths, not for score.

It takes as inputs first the height and width of the rectangle, then the individual characters of the rectangle separated by spaces. Example usage: To solve the rectangle

xxxyy xxyyy xyyxx 

you call the program like this: perl theprogram.pl 3 5 x x x y y x x y y y x y y x x

Anyway, by squaring the code like this, it should be possible to get down to score 0 or 2 with any programming language that has a one-line solution and a line comment character, by transforming the one-liner

one-liner 

like this:

##ooooooooooooooooooooo ##onnnnnnnnnnnnnnnnnnno ##oneeeeeeeeeeeeeeeeeno ##one---------------eno ##one-lllllllllllll-eno ##one-liiiiiiiiiiil-eno ##one-linnnnnnnnnil-eno ##one-lineeeeeeenil-eno ##one-linerrrrrenil-eno ##one-liner###renil-eno one-liner###renil-eno ##one-liner###renil-eno ##one-linerrrrrenil-eno ##one-lineeeeeeenil-eno ##one-linnnnnnnnnil-eno ##one-liiiiiiiiiiil-eno ##one-lllllllllllll-eno ##one---------------eno ##oneeeeeeeeeeeeeeeeeno ##onnnnnnnnnnnnnnnnnnno ##ooooooooooooooooooooo 

The only characters not part of a square are the two spaces. Depending on the language, there may also be ways to use clever multiline strings or comments (like Perl's here-docs) or some sort of redundant code to make those two space characters into a square.

As for getting the byte count low, my solution has a bunch of immediate improvements, and I'm sure there are also lots of other ways to refine this technique, but I don't have the patience for that; I just wanted to provide a demonstration so you guys can go wild with it.

MATL, 18 16 14 10 characters not in squares (92 103 111 127 bytes)

!!!ffqqqqQQQ%OOOOGGGxx44---@@&&lllxxTT ~~~ ++||7 ss--~~~ssvvss !!!ffqqqqQQQ"OOOOGGGxx44---@@&&lllxxTTYa~~~Z++||7Mss--~~~ssvvss 

Input is a character matrix with characters + and -. Doesn't count diagonal squares.

Try it online!

Or verify all examples and test cases.

Jelly,  20 1917 1908  1905 bytes, Score  20  0

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A full program taking a list of lists of . and + characters. (Works for any characters, even more than 2).

Try it online!

Any attempt to reduce score by introducing characters seems to just increase the score more. I guess the way to win this challenge is to find the right Lenguage!

Since Magma posted a very nice Perl solution scoring \$0\$ I decided to attempt similar using my \$20\$ byte solution:

ẆZṡLƊ€ẎṖƇZU$ƬḢ€ẎEƲ€S 

But Jelly has no comment character, so a little extra creativity is required...

Jelly programs consist of "Links" - code separated by newline characters, pilcrow characters (¶ - interpreted as the same byte as newlines) or valid code-page characters lacking a defined behaviour (at present q is an example). The latter actually count as a Link each rather than a pure separator. Program execution starts at the bottom-right most Link, the "Main Link" (here ⁽⁽ĿĿ).

We can call any other Link as a single-argument function (a "monad") by it's \$1\$-indexed position in the rest of the Links of the program using the "quick" Ŀ. These Links are indexable in a modular fashion; so if there are \$n\$ of them then the \$n^\text{th}\$ is the same as the \$0^\text{th}\$ and the \$-m^\text{th}\$ is the same as the \$(-m\mod n)^\text{th}\$.

So the program above starts by executing ⁽⁽ĿĿ, where ⁽⁽Ŀ is a representation of \$-27150\$ so ⁽⁽ĿĿ calls the \$-27150^\text{th}\$ Link which, since there are \$60\$ Links (due to the extra lines and the central nine qs), is the \$30 ^\text{th}\$ Link, ẆZṡLƊ€ẎṖƇZU$ƬḢ€ẎEƲ€S, to the left of the middle row of the \$3\$-by-\$3\$ square of qs. This is called with an argument of the program input:

ẆZṡLƊ€ẎṖƇZU$ƬḢ€ẎEƲ€S - Link: list of lists of characters, lines Ẇ - contiguous slices (i.e [[line1], [line2], ..., [lineN], [line1, line2], [line2, line3], ..., [lineN-1, lineN], [line1, line2, line3], ...., [lineN-2, lineN-1, lineN], ..., ..., lines]) € - for each (slice): Ɗ - last three links as a monad: Z - transpose - i.e. get the columns of the slice L - length - i.e. get the number of lines in the slice ṡ - overlapping slices of given length - i.e. get slices of columns of same length as lines in the original slice ...filled squares of all sizes (including 1-by-1) Ẏ - tighten - from a list of list of squares of each size to a single list of all squares Ƈ - filter keep if this is truthy: Ṗ - pop (truthy for all but the 1-by-1 squares) € - for each (filled square) Ʋ - last four links as a monad: Ƭ - collect up until results are no longer unique: $ - last two links as a monad: Z - transpose } U - upend } - together a quarter rotation Ḣ€ - head each - i.e. first row of each rotation Ẏ - tighten - i.e. get the entire perimeter (with repeated corners) E - all equal? (1 if so, 0 otherwise) S - sum