So, whoever wrote this post, are you, like, *proud* of it? Will you be proud of it in a decade? Will your children be proud that you composed, and publicly published, it?

That's one way to use the word "patriots", sure.

The sound of chair legs scraping against the floor in the cafeteria of our aquarium.

I got hungry. Don't shame me.

I love the mathematics of vote-counting! In university, I wrote IRV code that, for multi-seat elections, would distribute surplus votes in exact proportions, rather than taking a subset of the winner's ballots and using that as the surplus. For reproducability!

Isn't the plural, 'bi'?

Neat! Start with any parallelogram ABCD, pick a point on BC, call it E. ABCD can be transformed into AEFD by making the cut AE, and translating the triangle ABE so AB is along DC. This solution can be seen as two such transformations: get one edge to the right length, then, get a right angle.

If, after the first cut, you move the triangle to the right side of what remains of the rectangle, the third cut is just a continuation on the same straight line as the second cut.

Yes it is a different puzzle - at least, if you add the stipulation that flipping pieces over is undesirable. Use Andrew Stacey's diagram to cut one rectangle, then use its mirror image on the other rectangle. Mix and match, get two squares without flipping pieces.

Would the puzzle be different if you baked, say, 2 rectangles, and were able to mix & match, with the goal of making 2 identical squares of greatest area? If you baked N rectangles, with the goal of N squares?

The leftover rectangle is 1 1/3 by 7 2/3. Half the perimeter of the square is 20 2/3. 7 2/3 goes into 20 2/3 less than 3 times. Cut the leftover into 3 equal strips, and pad 2 sides of the square. - For what configurations would this algorithm terminate, I wonder?

Do rectilinear cuts make for 'better'? Cut a 9 by 1 1/3 strip off the short side, and another 9 by 1 1/3 strip again. You now have a 9 by 10 1/3 rectangle, and the makings of a long 1 1/3 strip to bring the 9 up to 10 1/3 too. Trim the end off the long strip.

You can reduce the area that's flipped with some extra cuts. Divide each right-angle triangle into an isocoles triangle, sharing the pointiest angle. This can be translated, only the piece at the shortest side of the original right-angle triangle will have to be flipped.

If you place two pieces next to each other, edge to edge, could you call it one 'cut' that's a straight line through both of them, through that shared segment? (Always looking for the edge cases!)

Ah, but where you have two conflicting requirements, you must give weights. Is an extra 0.1 inch on the side of the square worth 20 cuts? 2 cuts? Not even 1, it'd have to be an increase of at least 0.2 inches?