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This question is not the same as Adding coins inside a ring of coins

From a pile of equal size perfectly round coins take eighteen and make a perfect ring. Show that you can fit at least sixteen more coins inside the ring.

Not a solution because twelve is not sixteen.

Not a solution because twelve is not sixteen.

Please, do so without resorting to a computer, without "lateral thinking" and without referencing http://packomania.com/ or friends.

Note: The intended solution is pretty symmetric. The task is mostly about proving that it works.

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    $\begingroup$ I must say this does look similar to the linked puzzle... $\endgroup$
    – Dmitry Kamenetsky
    Commented Apr 23, 2021 at 12:46
  • $\begingroup$ So "perfect ring [of 18 identical round coins]" means "regular 18-gon". The generalization of the linked puzzle would be "What is the maximum number M of identical round coins that can be fitted inside a regular N-gon of coins? (without overlap and all lying flat)?" $\endgroup$
    – smci
    Commented Apr 23, 2021 at 20:56
  • $\begingroup$ @smci Unless OP of linked Q has something up their sleeve we all missed it is not answerable with reasonable effort let alone the generalization you propose. This Q, by contrast, has a clean and elegant geometric A. And on top of that I messed up and failed to exclude a second trivial-ish solution (even though I was aware of it). As I find it annoying when posters change their Q's to disallow legit A's others have put actual effort into I've fixed it in a new Q $\endgroup$
    – loopy walt
    Commented Apr 24, 2021 at 0:35
  • $\begingroup$ You can actually fit 19. $\endgroup$
    – Florian F
    Commented Apr 26, 2021 at 14:47
  • $\begingroup$ @FlorianF Very good! And it can still be validated elegantly and without computer! I've already posted two questions on this matter, so I feel I shouldn't post a third. But you could. (I can promise to not answer for a few days to give others a chance). $\endgroup$
    – loopy walt
    Commented Apr 27, 2021 at 13:54

2 Answers 2

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Not a perfect circle, but it is clear that it works, and I didn't use a computer:

enter image description here

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    $\begingroup$ "but it is clear that it works" Is it, though? It looks ok, but as we have seen looks may be deceptive. +1 for not using you-know-what, though ;-) $\endgroup$
    – loopy walt
    Commented Apr 23, 2021 at 15:23
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    $\begingroup$ very convincing. You have some mouldy coins :) $\endgroup$
    – Dmitry Kamenetsky
    Commented Apr 24, 2021 at 8:01
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    $\begingroup$ this looks like you are bragging you are rich. + 1 though $\endgroup$
    – Marius
    Commented Apr 26, 2021 at 7:51
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    $\begingroup$ @DmitryKamenetsky: he's a pirate! $\endgroup$
    – smci
    Commented Apr 26, 2021 at 19:02
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Let's calculate the radius of the ring of 18 coins, taking the radius of the coin as $1$.

The centres of the coins form a regular 18-gon. Two adjacent coin centres and the origin form an isosceles triangle with an angle at the origin of $360/18=20$ and base length $1$. We then get $$ r \sin \frac{20}2 = \frac12$$ So $r=2.879$. This is the distance from the origin to the centres of the outer ring of coins. Note that this radius was rounded down so that the area to be filled will be underestimated.

Now let's try to fill the inside area by

dividing it into rings or annuli of width 1, and filling those with coins. So one ring has its coins at distance $1.879$, and inside that is another ring with the coins at distance $0.879$ from the origin.
We now use the same formula as before but calculate the angle from the radius instead of the radius from the angle. $$ 1.879 \sin \frac{\alpha}{2} = \frac12\\ \alpha=30.9$$ This means we can fit $\lfloor \frac{360}{30.9} \rfloor = 11$ coins in the ring. Similarly for the inner ring we get $$ 0.879 \sin \frac{\alpha}{2} = \frac12\\ \alpha=69.3$$ This means we can fit $\lfloor \frac{360}{69.3} \rfloor = 5$ coins in the inner ring.

So it is fairly straightforward to fit 16 coins inside the ring of 18 coins.

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  • $\begingroup$ I can see I should have plugged this loophole more forcefully than by asking for something "pretty symmetric"... $\endgroup$
    – loopy walt
    Commented Apr 23, 2021 at 15:40
  • $\begingroup$ Could you perhaps make the trigonometry non-computery, so we can all move on? $\endgroup$
    – loopy walt
    Commented Apr 23, 2021 at 16:42
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    $\begingroup$ @loopywalt - calculation to 3 decimal places falls far short of being "computery", and most definitely does not violate your "no computers" rule (some of us were trained to do these sorts of calculations by hand - though in my case, it was outdated before I ever had to do it in earnest). This meets everything you asked for, even if it isn't what you wanted. $\endgroup$
    – Paul Sinclair
    Commented Apr 23, 2021 at 17:17
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    $\begingroup$ @PaulSinclair I do not really have any strong feelings on that but the tag def says "A puzzle designed to be solved without using calculators,[...]". At the very least this means that things are nowhere near as clear cut as you claim. I find it difficult to see the point of disallowing calculators if numerical evaluation of transcendental functions were considered paper-and-pencil matter. $\endgroup$
    – loopy walt
    Commented Apr 23, 2021 at 18:06

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