"You're planning the seating for a major round-table event. The client keeps adding rules, and every miscount means the wrong number of place settings gets ordered. This time, a number alone won't be enough: you'll need to convince the client your formula is always right, not just right for the case you happened to check."
List every way to arrange 3 people (A, B, C) in a straight line.
Wrap each line-up into a circle. Do any describe the exact same circular seating?
How many different circular seatings for 3 people? Find a general formula for \(n\) people.
Explain why your formula works for any \(n\), not just the cases you checked.
Now the seating can be viewed from the other side (like a bracelet), so a mirror image counts as the same seating. How must your formula change?
Prove your new formula holds for any \(n\).
Normal circular table, 6 guests. A and B insist on sitting together. How many valid seatings?
If \(k\) specific guests must all sit together among \(n\) guests, find a formula in \(n\) and \(k\) and justify it.
8 guests. A and B must sit together, but C and D refuse to sit next to each other. How many valid seatings satisfy both conditions?
Explain your approach and name the earlier phases' ideas you combined.
Add a third pair, E and F, who also refuse to sit together. Extend your method to handle all three conditions.