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we have 10 coins, and we aligned 9 coins as shown below. The extra coin shown is being used to turn around the aligned coins.

enter image description here

By turning it around like shown below, how many times would the coin turn around when it comes to its original position?

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    $\begingroup$ Note: I don't find it annoying, but I have noticed that in the past some users have been irritated by other users posting lots of questions within a short time span. You don't have to, but I would propose slowing down just to be on the safe side. $\endgroup$
    – boboquack
    Nov 10, 2017 at 10:22
  • $\begingroup$ You don't have to apologise to me, I'm just giving advice from my experience on the site. As I said, I don't really care. $\endgroup$
    – boboquack
    Nov 10, 2017 at 10:25
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    $\begingroup$ Suppose you had one coin fixed and you rolled another coin around it 360 degrees. Would you consider the coin to have rolled "1 turn" or "2 turns"? On the one hand, the coin rotated 360 degrees as it traveled. On the other hand, it also traveled in a circle, which adds another 360 degrees. So from the reference point of the moving coin it rolled once but from the reference point of fixed space it rolled twice. $\endgroup$
    – JS1
    Nov 10, 2017 at 11:24
  • $\begingroup$ @JS1 yes, technically it rotates twice even though it passes every point in the coin only once while turning around. $\endgroup$
    – Oray
    Nov 10, 2017 at 12:20
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    $\begingroup$ I kind of feel cheated, seeing that the tick went to the answer that still has mistaken logic in the explanation, and the correct answer was edited in only about an hour after I posted the correct solution. $\endgroup$
    – Bass
    Nov 10, 2017 at 16:40

4 Answers 4

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Since we have symmetry, let's calculate the arc touched while traversing a quarter of the distance, for example, from the 6 o'clock position to the 9 o'clock position.

We get 30 + (30 + 90 + 30) + 30 = 210 degrees.

Multiply that by four to get the total arc length of

840 degrees. Since the rotation of the coin is twice the arc traversed, we get a grand total of 1680 degrees, or 4⅔ full rotations.

Tested by rolling some poker chips around each other, they had six markings along their perimeter, which seemed to align just about exactly right.

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It is

$4\frac{2}{3}$times

Calculations:

enter image description here

As mentioned by JS1 and ffao, I had missed the extra one rotation of the outer coin.

In terms of circular distance,

$2\pi R \implies 2$ rotations
$\frac{14}{3}\pi R \implies \frac{14}{3}*\frac{2}{2}$ rotations
$\frac{14}{3}$ rotations.

Please ignore the $\frac{22}{7}$ part :)

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    $\begingroup$ But does a mixed fraction make better sense? Some would disagree... $\endgroup$
    – boboquack
    Nov 10, 2017 at 10:37
  • $\begingroup$ @boboquack A good question. I think it would make better sense. Saying that a coin rolled twice and 1/3 part more, makes more meaning than 7/3. $\endgroup$
    – ABcDexter
    Nov 10, 2017 at 10:39
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    $\begingroup$ You missed one rotation, see JS1's comment on the question or Seyed's link to understand why. $\endgroup$
    – ffao
    Nov 10, 2017 at 14:09
  • $\begingroup$ @ffao Aah, my bad. $\endgroup$
    – ABcDexter
    Nov 10, 2017 at 14:14
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    $\begingroup$ @ffao is right. The total length of the perimeter of the 9 coins that the rolling coins touches is 7/3 of a coin circumference. So if that rolling coin rolled along a straight line for that distance, then your answer would be correct. $\endgroup$
    – Jaap Scherphuis
    Nov 10, 2017 at 14:14
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If a coin rotates around another coin, it completes two full rotation. https://en.wikipedia.org/wiki/Coin_rotation_paradox enter image description here enter image description here

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    $\begingroup$ The description of the corner coin travel (five twelfths) is correct, but then you switch to seven twelfths in the equation, which is incorrect. The drawing also shows the incorrect angle. $\endgroup$
    – glibdud
    Nov 10, 2017 at 16:29
  • $\begingroup$ @glibdud, you are right, thanks for mention it. I edited the picture. $\endgroup$
    – Seyed
    Nov 10, 2017 at 17:03
  • $\begingroup$ I think you went in the wrong direction... the five twelfths was correct. $\endgroup$
    – glibdud
    Nov 10, 2017 at 18:26
  • $\begingroup$ @glibdud, I don't know what I was thinking. You are right it was five twelfths. I edited it again although it is too late :) $\endgroup$
    – Seyed
    Nov 10, 2017 at 19:06
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It seems it has to subtend

angles of 60 degrees at the 4 center coins and 120 degrees at the 4 corner coins, there by resulting in a total of 240 + 480 = 720 degrees, which translates to 2 complete turns

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    $\begingroup$ To me it looks like 150 degrees around the corners. $\endgroup$
    – JS1
    Nov 10, 2017 at 11:19
  • $\begingroup$ It is 150 degrees around the corners. $\endgroup$
    – ABcDexter
    Nov 10, 2017 at 11:20

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