Three squares. The top line segment is variable in length. What’s the minimal blue fraction?

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## Solution

The minimal blue fraction is 1/5.

For convenience we set the square side to 1. From similarity of the two triangles formed by the oblique lines, we can calculate the lower square area and multiply by x^{2} to obtain the upper area. Also, we see that the sum of the heights must be 1, so h=1/(1+x).

From similarity of triangles ABO and HGO, we deduce h/1=(h-a)/a, leading to a=h/(1+h)=1/(2+x). So for the blue fraction we get A(x)=(1+x^{2})/(2+x)^{2}.

To find the minimum of this function we take the derivative and set it to 0. This gives us x=1/2, corresponding to a fraction of 1/5, and this is indeed a minimum on the range [0,1].

## Poem

3 squares

Red, yellow and blue

What can they do?

What is their meaning?

Just staying here in this figure

A draft for a flag?

The red colour for blood

From our soldiers

Blue for truth wisdom

A symbol for a city, a country

A flag with pride

Nobody will hide

## 2 replies on “Who’s afraid of red, yellow and blue?”

Interesting problem! After reading your solution, at least I understood how it is done. The challenge in problems like this is always to reduce the variable parameters to one, in order to get the derivative, and I think, it requires a lot of practice (which I never had) to “see” the path.

Thanks. You’re right about the strategy, although I think with multiple variables extremes can also be found. In any case, it involves differential calculus.