cs-020 — The Josephus Ring
| Field | Value |
|---|---|
| id | cs-020 |
| difficulty | Advanced |
| topics | simulation, math, recurrence |
| lecture | lecture-05 |
| estimated time | 5-minute attempt + 10-minute discussion |
| prerequisites | functions, dictionaries, and earlier algorithmic patterns |
| objective | design the algorithm, verify it on edges, and argue correctness or trade-offs |
| solution | held by the instructor; discussed after the attempt |
Problem
People stand in a circle; every k-th is removed until one remains. Derive the survivor with a recurrence, verify by simulation. A first taste of proof plus experiment.
Your deliverable is the stated assumptions, the survivor count for the worked instance, the counting rule in one sentence, and the wrap boundary (who is counted when the circle closes) traced. If the problem is ambiguous anywhere, state the assumption you are making and carry on — a stated assumption is part of a good answer.
Examples
- Input: n = 7 people, k = 3
- Output: survivor position 4 (0-indexed 3)
Solve for the worked instance first. Then ask what would change if one number, name, or condition were replaced by something stranger.
What to notice
- Recurrence on the circle size: J(1)=0; J(n) = (J(n-1)+k) mod n. Prove it by drawing n=5.
- Writing the rules IS the solution; code is transcription.
- Every assumption is a place where two good answers can differ.
Discussion starter
After the solution is revealed, the class spends five minutes on this prompt: "Which assumption would you attack first if you had to argue the other side?"
Constraints and assumptions
- Difficulty: Advanced (functions, dictionaries, and algorithmic reasoning assumed).
- Topics: simulation, math, recurrence.
- Classroom-scale inputs unless the problem says otherwise.
- Time-box: about five minutes for a defensible plan; a complete solution may take longer and that is expected - state where you would resume.
- Optional stretch (not required): describe an alternative approach and say when it beats the obvious one.
Hints and extension
- First nudge: identify the pattern the worked instance uses before computing anything.
- If stuck after three minutes, write the input/output line you expect and trace the smallest example by hand.
- After the reveal, attempt the extension below before the next lecture.
- Extension challenge: Find the closed-form answer and prove it agrees with your loop.