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?
(Figure: in case M the four circles sit in a 2 by 2 arrangement and in case N the nine circles in a 3 by 3 arrangement; in each case the circles touch one another and the sides of the square.)
Answer: B
The ratio is 1 : 1.
In case M each circle has diameter 1/2 cm, and in case N each has diameter 1/3 cm. The total shaded area is
\[ \begin{aligned} M &: 4 \times \pi \left(\tfrac{1}{4}\right)^{2} = \frac{\pi}{4} \\ N &: 9 \times \pi \left(\tfrac{1}{6}\right)^{2} = \frac{\pi}{4} \end{aligned} \]
Both shade \(\pi/4\) of the 1 cm2 sheet, so both leave \(1 - \pi/4\) unshaded, and the ratio is 1 : 1. In general, n2 touching circles in an n by n grid always cover \(\pi/4\) of the square. The options 2 : 3 and 3 : 2 come from comparing the numbers of circles instead of their areas.
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