# Mathematical beauty

Reading science books for the general public, you’ll often find physicists talking about elegance, beauty and words of the like describing laws or theories.

The Wikipedia has an entry for “Mathematical Beauty”. Another entry says “Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof […]”.

The Spanish journal El Pais is publishing each week a mathematical challenge to its readers to commemorate the 100th anniversary of the Spanish Royal Mathematics Society.

Last week’s challenge was to solve the sides of the different inner squares that compose the following rectangle, knowing that the red one has a side of 3.

Rectangle made of squares (the squares are deformed in the image).

You may argue that the inner figures are not squares but rectangles; this is because the sizes have been disguised to hide the solution to the problem. The relative positions however have not been twisted. You may read the explanation here.

How did I approach the problem?

I used as many unknown variables as needed, and related them with as many equations as needed, ending up with a linear system of 14 equations. Not difficult to solve with Excel. You may see the process and the solution below.

Rectangle with 14 variables

My approach to the problem...

When I watched the solution proposed by the mathematician I had an “aha” moment. I immediately recognized the beauty in the way she solved it. You may see a video with her solution in the here (in Spanish – 6 minutes).

Instead of filling the problem with variables and equations, as I did, she reduced them to really the minimum needed: 2 variables (x and y)  and 2 equations:

(3x -y + 3) + x = 3 + (2x + y + 12) or x = y + 6

(10y – x + 3) + (4y – x) = (3x -y + 3) + (2x – y + 3)

I guess that not being a mathematician I did not care much about the simplicity I employed, knowing that the tools I counted with would not have had any problem in dealing with the calculation… a different mindset :-).

Solution.