"Cut into halves" vs. "cut in half"

Question

1. It’s easy to cut a rectangle into halves.
2. This will cut both rectangles in half.

What is the difference between halves and half?

There are two answers, and it’s best to get both. "It’s easy to cut a rectangle into halves." All you have to do is make sure that the cut passes through the precise center of the rectangle. The cut can be at any angle. Here we have two rectangles, a positive one (the cake) and a negative one (the missing piece). Decide where the centers of the two rectangles are. These two points make a line. Make a cut along that line, and the cake will be split evenly. "This will cut both rectangles in half", so each piece will equal half of the cake plus half of the missing piece. In other words, both of the resulting pieces have the same area.

Answer 1

The bulk of your question seems to be quoting an answer to an interview question about cutting a rectangular cake into two equal pieces when a rectangular piece has already been taken. For the first bolded item, “It’s easy to cut a rectangle into halves”, one could just as well say “It’s easy to cut a rectangle in half”; and similarly for “This will cut both rectangles in half”, one can say “This will cut both rectangles in halves”. In the particular cases in this example, either form can be used; by habit I'd probably use half.

The term halves refers to the parts that result when something is cut in half. In the interview problem, the important thing to emphasize is that each rectangle is cut in half, or sliced into two equal-sized parts. You can emphasize that via either wording.