https://fivethirtyeight.com/features/can-you-solve-these-colorful-puzzles/
Looks like answer is:
Distinct Balls:
- 2: 1 turn (base case)
- 3: 4 turns
- 4: 9 turns (example in problem)
- 5: 16 turns
- 6: 25 turns
- . . .
You see the pattern. For n
distinct balls, on average, it takes (n - 1)^2
turns before the game stops.