Some perfectly circular coins are placed in a nearly circular chain. Each coin touches two other coins in the chain. There is room for least one more coin inside the chain.
A new coin is placed inside the chain and rolled around the chain once completely. The rolling coin touches at least one coin in the chain throughout the roll. It rolls without slipping. Then this coin is placed outside the chain and rolled around it in a similar manner in the opposite direction. All coins have the same radius. There is enough room for the rolling coin to touch each coin at least once in each roll.
How many times should we perform these two rolls so that the number of rotations made by the coin while rolling is twice the number of coins in the chain?