- Blank A4 sheet of paper; and
- A ruler (also to use as a straightedge).
- Draw a square with a side-length of $a=16$cm.
Then, draw eight circles inside. The circles must have $2$cm in diameter; they must all be the same size; they must not touch each other; their distances must all be different from each other (i.e. their positions must be randomised); and they must not touch the perimeter of the square. $$\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc$$
Now, draw two points on each side of the square. Begin with the top side, and from left to right, make points $A_1$ and $B_1$. The distance from these two points is arbitrary, but none of them can be on the corners of the square. Now, rotate the square $90^\circ$ anti-clockwise. You will now have a new top side, where you can put points $A_2$ and $B_2$ from left to right. Their distance is also arbitrary and cannot be on the corners of the square, but it also cannot be the same distance as any other points placed (i.e. the previously placed points).
Continue with this method to make points $A_3$ and $B_3$, and then $A_4$ and $B_4$, making sure that the distance between each two points on a side is unique. Now, connect the points with a line in the following fashion. $$A_1\to A_2\to A_3\to A_4\to B_1\to B_2\to B_3\to B_4\to A_1$$
But, ensure that the lines do not touch nor intersect any of the circles!
Note: You might have to place your points carefully.
$$\verb|Ensure the lines do not touch nor intersect any of the circles!|$$
Removed Part 2 of the puzzle as it is much too difficult and incorrect (or perhaps impossible) in very specific cases of drawing circles. Thus, I am adding a bonus:
Bonus: What is the minimum value of $a$ for your specific case you have chosen (i.e. how you have randomly positioned the circles)? Note that you cannot move the circles in different positions after plotting them.
I used the tag
connections-puzzle because you have to connect points with lines.