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   Covering square with 2x1 rectangles
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   Author  Topic: Covering square with 2x1 rectangles  (Read 185 times)
rmsgrey
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Covering square with 2x1 rectangles  
« on: Aug 9th, 2021, 4:41am »
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"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?
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Grimbal
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Re: Covering square with 2x1 rectangles   Split_squares.png
« Reply #1 on: Aug 10th, 2021, 1:46pm »
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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.
« Last Edit: Aug 11th, 2021, 1:03pm by Grimbal » IP Logged

rmsgrey
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Re: Covering square with 2x1 rectangles  
« Reply #2 on: Aug 11th, 2021, 6:52am »
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Can you prove the impossibility of 1, 3 and 4?
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Grimbal
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Re: Covering square with 2x1 rectangles  
« Reply #3 on: Aug 11th, 2021, 12:54pm »
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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.
« Last Edit: Aug 11th, 2021, 1:01pm by Grimbal » IP Logged
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