Infinite Square Inscription

Geometry Level 3

Squares are repeatedly inscribed at an angle of $45^{\circ}$ within other squares. The result of the first six steps of this process is shown below.

If this process is performed infinitely many times, what would the area shaded blue be, as a fraction of the area of the outermost square?

\frac{3}{5}

\frac{5}{8}

\frac{3}{4}

\frac{7}{8}

\frac{5}{6}

\frac{4}{5}

\frac{1}{2}

\frac{2}{3}

1 solution

Zane Gates
Jun 7, 2019

First, draw in horizontal and vertical lines to divide the figure into four quadrants. The four outer blue triangles together constitute exactly half of the figure.

Next, consider the three triangles dotted red. They are clearly the same size, and one out of the three is shaded blue. This pattern of three triangles repeats inwards: the three yellow triangles are of the same size, and one out of three is blue. It also repeats in each of the four quadrants: ditto, with the green dots. Of this half of the total area, one third of it is blue.

So, the total area shaded blue is $\frac{1}{2} + (\frac{1}{3} \times \frac{1}{2}) = \frac{3}{6} + \frac{1}{6} = \frac{2}{3}$ .

Infinite Square Inscription

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