Dear Uncle Colin,

I have been presented with this diagram: the marked points are a third of the way along the sides. I need to know what fraction of the grey square is shaded blue. Any ideas? A square in a square

Somehow Hued Area Defies Evaluation

Hi, SHADE, and thanks for your message!

There are several ways of doing this, some brutal and some more elegant.

Let’s do it a brutal way first.


Suppose the coordinates of the corners of the square are $(0,0)$, $(0,1)$, $(1,1)$ and $(0,1)$.

Then an equation of the line through the origin is $y=3x$, and an equation of the line through the top-left corner is $3y + x = 3$.

Substituting into the second gives us $10x = 3$, so one of the crossing points is at $(0.3, 0.9)$.

There are at least two ways to go here: symmetry can give us the other coordinates and we can Pythagoras the whole thing up. That’s a perfectly reasonable way to get the answer if you’re a computer scientist or something, but around here we value a little bit of elegance.

Instead of looking at the square, we can look at the triangle with its apex at $(0.3, 0.9)$ and its base along the left-hand side of the square. It has a ‘height’ (measured horizontally) of 0.3 and a base (vertically) of 1, so its area is 0.15.

That triangle is repeated four times around the edge of the blue square, making a total of 0.6 units squared, and the area of the square is therefore 0.4.


An elegant decomposition

If I add in more parallel lines, splitting the shaded bit into four equal pieces, and jigsaw the outside pieces around to make congruent squares, I find there are six of those. So the blue area is two-fifths of the shaded area.

Hope that helps!