Nested shaded squares

This problem is a classic from the HSC Sequences and Series topic, often appearing in the context of infinite geometric progressions. By chasing the pattern of shaded quarters, you’ll see how a simple visual construction gives rise to a neat sum‑to‑infinity calculation—a skill that will serve you well in both extension courses.

Problem Statement

In a unit square, shade the bottom-left quarter. Then repeatedly subdivide the top-right quarter into four equal squares and shade its bottom-left quarter. Find the total shaded fraction.

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Hints

Shaded areas form an infinite GP with first term $\frac14$ and ratio $\frac14$.

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Solutions

Here, the shaded areas form an infinite geometric progression. The first shaded square occupies the bottom‑left quarter of the unit square, so its area is $a = \frac14$. After shading this, we focus on the top‑right quarter of the original square, which is itself a $\frac14 \times \frac14$ square. Subdividing it into four equal squares and shading its bottom‑left quarter gives a shaded area of $\frac14 \times \frac14 = \frac{1}{4^2}$. Each new shaded region is a quarter of the previous one, so the common ratio is $r = \frac14$.

Because $|r| < 1$, we can apply the sum‑to‑infinity formula $S_\infty = \frac{a}{1-r}$:

$$S_\infty=\frac{\frac14}{1-\frac14}=\frac13.$$

Therefore, the total shaded fraction is $\frac13$.

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Takeaways

• In these patterns, look for a constant scaling factor; here each new shaded square is $\frac14$ of the previous one.
• The sum‑to‑infinity formula $S_\infty = \frac{a}{1-r}$ applies whenever $|r|<1$, giving a finite total area from infinitely many pieces.
• Translating a visual subdivision into a geometric progression quickly yields the sum.

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Further Readings

HSC Mechanics, HSC Trigonometry, HSC Vectors, HSC Combinatorics