13. A number is written in each of the twelve circles shown. The number inside each square indicates the product of the numbers at its four vertices. What is the product of the numbers in the eight grey circles?
A 20
B 40
C 80
D 120
E 480
Solution
B

If the products inside the squares are $P, Q$, $R, S, T$, then the required product is $\frac{P Q S T}{R^{2}}$. This is because the numbers in grey circles are included once each in the products $P, Q$, $S$ and $T$ and those in white circles show up twice each. So the product of the numbers in the grey circles is $\frac{10 \times 4 \times 6 \times 24}{12 \times 12}=10 \times 2 \times$ $2=40$. The picture on the right shows an example with a possible construction.