Intuition usually imagines that a coin going around an identical coin makes one full turn. But as it rolls, it does not only move forward: it also rotates along the path it travels.
The Coin That Goes Around Another Coin
Riddle statement
You have two identical coins on a table.
One stays fixed.
The other rolls around the first, without slipping, until it returns to its starting point.
How many complete turns has the moving coin made around its own center?
Show solution
Solution
The answer is:
complete turns.
The quick intuition is usually 1, because the moving coin goes around an identical coin.
But we must look at the path traveled by the center of the moving coin.
If each coin has radius $r$, the center of the fixed coin is at distance 0 from itself. But the center of the moving coin, as it rolls around, is always at distance:
from the center of the fixed coin.
Therefore, the center of the moving coin traces a circle of radius $2r$, whose circumference is:
The circumference of the moving coin itself is:
Each time the coin advances a distance equal to its own circumference, it makes one complete turn.
Since it travels:
and each turn corresponds to:
it makes:
complete turns.
Answer: it makes 2 complete turns.