Squaring the square

A

apple2357

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Mar 9, 2018
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Consider this problem...

1652264806624.png

Its relatively easy to get started and i have got this far:

1652264968905.png

The tiny square is obv 2, but what next? Algebra or is there an additional insight someone can suggest? I think its a famous problem so the solution is out there but more interested in knowing what to do!
 
apple2357 said:
Consider this problem...

View attachment 32691

Its relatively easy to get started and i have got this far:

View attachment 32692

The tiny square is obv 2, but what next? Algebra or is there an additional insight someone can suggest? I think its a famous problem so the solution is out there but more interested in knowing what to do!
Click to expand...
I would certainly use algebra.

Label a few squares with variables or expressions, and write equations you can solve. For example, if the lower right square is x, what is the upper left (based on the fact that the whole figure is a square)? Write those in, and then fill in others either with expressions in x, or a new variable.
 
The side of the tiniest square is 2 = 9 - 7.

I suspect that the best way to attack this is to expand the picture and label line segments with variables representing length because I get confused just looking at this diagram. At the end, I suspect you will use the fact that the sums along all four sides of the macro square are equal.
 
I have done this before, but with 3 given squares. I'm not sure that it's possible to do it with only 2.
mathworld.wolfram.com

Perfect Square Dissection -- from Wolfram MathWorld

A square which can be dissected into a number of smaller squares with no two equal is called a perfect square dissection (or a squared square). Square dissections in which the squares need not be different sizes are called Mrs. Perkins's quilts. If no subset of the squares forms a rectangle...
mathworld.wolfram.com
Scroll down to the animation.
:)
 
I've worked this out with algebra. I recommend using the following 4 variables a,b,c & d...

sq.png

1. Horizontally, write an alternative expression for the length of the top of square "a" giving a = <expression in terms of b,c>
2. For the "whole" outer square, the horizontal size (in terms of c,b,d) = vertical size (in terms of c,b,d). This gives an equation relating d&c
3. Bottom right square, horizontal size (in terms of a,b) = vertical size (in terms of a)
4. Horizontally for the top of square "a+b" write a+b = <expression in terms of c,d,b>

These 4 equations provide enough info to calculate values for a,b,c and d. Then it's possible to answer the original question
 
Many thanks. I did start using algebra but tried to keep it down to two variables and kept getting lost. But I see you went to four and didn’t cause too many issues. Well done !
 
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