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        riddles >> medium >> Covering square with 2x1 rectangles
        
(Message started by: rmsgrey on Aug 9^th, 2021, 4:41am)

Title: Covering square with 2x1 rectangles
Post by rmsgrey on Aug 9^th, 2021, 4:41am

"Borrowed" from a recent Numberphile video:

Given a square, it's possible to divide it into some number of similar rectangles, each with sides in ratio 2:1. What numbers of rectangles are possible, and what numbers are impossible?

Title: Re: Covering square with 2x1 rectangles
Post by Grimbal on Aug 10^th, 2021, 1:46pm

You can do any value except 1, 3 and 4.

2 is obvious.

For every N, you can make a new square with N+4 rectangles by surrounding the square by 4 rectangles. See 2 -> 6 for example.

For every N, you can make a new square with N+3 rectangles by splitting a rectangle in four. See 2 -> 5 for example.

7 is actually a 2 with one rectangle split in 9, giving 10, then four rectangles joined to obtain 7.

From 5, 6 and 7 you can construct any N > 7.

Title: Re: Covering square with 2x1 rectangles
Post by rmsgrey on Aug 11^th, 2021, 6:52am

Can you prove the impossibility of 1, 3 and 4?

Title: Re: Covering square with 2x1 rectangles
Post by Grimbal on Aug 11^th, 2021, 12:54pm

1 rectangle: obvious.

I.e. the square must be covered by one rectangle. That rectangle must have area 1 and ratio 2:1. The rectangle must have dimensions sqrt(2) by sqrt(0.5). The diagonal of the rectangle is sqrt(2+0.5) = sqrt(2.5). This diagonal doesn't fit in the square with diagonal sqrt(2). So a rectangle with sufficient area doesn't fit in the square.

(Or more simpler: the square isn't a 2:1 rectangle.)

3 rectangles:
There are 4 corners and only 3 rectangles. One rectangle must fill two corners. That rectangle must have size is 1 x 0.5. The remaining area is also a 1 x 0.5 rectangle. And it must be filled by wo rectangles. Again, one rectangle must fill 2 corners of that area. This leaves only 2 ways, 1 x 0.5 or 0.5 x 0.25. One way doesn't give space for a third rectangle, the other leaves a 3:2 space which is not the right ratio.

4 rectangles:
If one rectangle fills 2 corners, there is always one rectangle that must fill a whole side. That leaves few possibilities.
The first rectangle would be 1 x 1/2. The second can only be 1/2 x 1/4. The third can be 1/2 x 1/4 leaving a 1/2 x 1/2 space or 3/4 x 3/8 leaving a narrow 3/4 x 1/8 rectangle. Both are the wrong ratio.

So remains the case where each corner of the square is filled by a different rectangle.

In that case, consider a rectangle that touches the center of the square. There must be one. That rectangle must reach from a corner to the center. It must have at least 1/2 in both dimensions, which actually means it must have dimensions 1 x 1/2. We are back to the previous case which didn't work. (Or in fact we contradict the hypothesis that each corner is filled by a different rectangle).

I think that covers it. No pun intended.