4. In how many ways can the line $x \sin \sqrt{16-x^{2}-y^{2}}=0$ be drawn without lifting the pencil and without retracing any part of the line? (12 points)
Solution. Since $\pi^{2}<16<(2 \pi)^{2}$, the given line consists of 2 circles with radii 4 and $\sqrt{16-\pi^{2}}$ and a vertical segment.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e353629715b7c8c09e0g-08.jpg?height=214&width=214&top_left_y=1115&top_left_x=955)

This line is unicursal, as it has only 2 odd points $\mathrm{A}(0 ; 4)$ and $\mathrm{B}(0 ;-4)$. You should start drawing the line from one of these points and finish at the other (2 options). Next, there are 3! ways to choose the order in which to pass the left arc, the right arc, and the diameter. When passing the diameter, there are also 3! ways to choose the order in which to pass the arcs and the diameter of the inner circle. In total, the number of ways is $2 * 3! * 3! = 72$.

Answer: 72.