Discussion
Equal sized circular regions are shaded in a square sheet of paper of 1 cm side length. Two cases, case M and case N, are considered as shown in the figures below. In the case M, four circles are shaded in the square sheet and in the case N, nine circles are shaded in the square sheet as shown.
What is the ratio of the areas of unshaded regions of case M to that of case N?
Answer: B
The ratio is 1 : 1.
In case M, four circles fill the 1 cm square in a 2 by 2 grid, so each has diameter 1/2 cm and radius 1/4 cm. In case N, nine circles fill it in a 3 by 3 grid, so each has radius 1/6 cm. The shaded areas are:
\[ \begin{aligned} M: \; 4 \times \pi \left(\tfrac{1}{4}\right)^{2} &= \tfrac{\pi}{4} \\ N: \; 9 \times \pi \left(\tfrac{1}{6}\right)^{2} &= \tfrac{\pi}{4} \end{aligned} \]
Both unshaded areas are therefore \(1 - \pi/4\) square cm, and the ratio is 1 : 1.
In general, \(n^{2}\) touching circles in an \(n\) by \(n\) grid always cover \(\pi/4\) of the square, whatever \(n\) is. The ratios 2 : 3 and 3 : 2 come from comparing the numbers of circles, 4 and 9, or their rows, instead of their areas.
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