How many turns does the small circle make?
A hard-level "Count" problem.A rolling circle's turn count comes from the path of its centre, not the length it touched. Outside adds one turn; inside takes one away.
GivenA circle of radius 1 cm rolls once round the outside of a circle of radius 3 cm without slipping / π = 3.14
Hints
- The natural first attempt: the distance it rolls along is the big circle's circumference, 18.84 cm, and the small circle's own is 6.28 cm. Dividing gives 3 turns. But this is wrong.
- Place the small circle at several positions and mark only its centre. Those centres trace a circle one step outside the big one. That path — not the contact line — is what sets the number of turns.
- The centre's path is a circle of radius 3 + 1 = 4 cm — bulged outwards by the small circle's own radius.
- So it is centre path 25.12 cm ÷ own circumference 6.28 cm, which reduces to (3 + 1) ÷ 1 = 4 turns — exactly one more than the first attempt.
- The extra turn comes from the fact that going once round the big circle also turns the small circle once in space. Rolling on the inside, that rotation reverses, so you get one fewer.
Answer 4
More "Count" problems
- How many cubes have 2 painted faces?
- Into how many pieces do 4 lines cut the paper?
- In how many ways can the three chosen rods form a triangle?
- How many triangles are there in the figure?
- How many squares are there altogether?
- How many squares does the diagonal pass through?