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