Problem 3. For her birthday, Princess Arabella received a ring adorned with a precious stone. The stone was in the shape of a rectangular prism, such that the areas of its three sides were $1 \mathrm{~mm}^{2}, 3 \mathrm{~mm}^{2}, 12 \mathrm{~mm}^{2}$ respectively. How much does the stone weigh, given that $1 \mathrm{~mm}^{3}$ of it weighs 0.003 grams?
Solution. Let's denote them as $a, b$, with the lengths of the three sides of the stone. From the condition of the problem, we have that $a \cdot b=1, b \cdot c=3$ and $a \cdot c=12$. Using these equalities, we get,

$$
1 \cdot 3 \cdot 12=(a b) \cdot(b c) \cdot(c a)=(a b c) \cdot(a b c)=V \cdot V=V^{2}
$$

where $V$ is the volume of the stone, so from this we get that the volume of the precious stone is $V=\sqrt{1 \cdot 3 \cdot 12}=\sqrt{36}=6 \mathrm{~mm}^{3}$. Finally, the desired weight of the stone is $6 \cdot 0.003=0.018$ grams.