$\verb|Eight Circles|$

Question

^{An entry in Fortnightly Topic Challenge #37: Rare and Endangered 1}

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Materials

• Pencil;
• Eraser;
• Blank A4 sheet of paper; and
• A ruler (also to use as a straightedge).

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Puzzle:

1. Draw a square with a side-length of $a=16$cm.

2. 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. $$\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc\quad\bigcirc$$

3. 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.

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Aim:

$$\verb|Ensure the lines do not touch nor intersect any of the circles!|$$

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Edit:

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.}

Answer 1

Maybe I am misunderstanding the puzzle, but it seemed straightforward to

take the requirement of not touching the circles into account when placing the points in order.

I literally plopped down circles as random as I could (but aligned to a grid, for ease of working) in a convenient drawing tool, in a 16x16 square. Then I

added the points while tracing the line in order $A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4, A_1$, ensuring the lines do not intersect circles along the way. I moved the first point when it turned out to be positioned in an inconvient spot. Finally I ensured distances to be unique.

The result is:

As $a = 16$, I wonder if we are allowed to break that requirement for the bonus? Of course, circle placement will not be random anymore when the square barely provides enough room to hold them.

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I allowed myself to go all out and move the circles and the points while trying to make everything fit. It then was not hard to construct the below example, which matches all the requirements in an 8x8 square. Of course, this is not in the spirit of the puzzle, but it shows that obtaining $a < 16$ for the bonus question is possible if one places the circles lucky.

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Side note: when looking at this puzzle from a mathematical perspective, one can place the points arbitrarily close to the corners (but with sliiiightly different distances to fullfill the requirements). The lines are then arbitrarily close to the sides of the square, transforming this in a question of packing 8 circles with diameter 2 in a square that is as small as possible. According to Wikipedia, that can be done in a square with sides $2 + \sqrt{2} + \sqrt{6} \approx 5.863\ldots$, so let's say 6.

You can imagine that the blue lines can move arbitrarily close to the side of the square; in the limit, the square will have side lengths $2 + \sqrt{2} + \sqrt{6}$ as mentioned above.