Circular Review Table: Testing a Seating Constraint Set
Six project team members-Anita, Bala, Chetan, Divya, Esha and Farhan-must be seated around a circular review table for a client presentation. All face the centre, and the seats are equally spaced. The team lead has fixed five conditions for the name cards. Anita must sit opposite Divya. Bala must sit second to the left of Chetan; here 'second to the left' means two seats in the clockwise direction. Esha must not sit adjacent to Divya. Farhan must sit adjacent to Anita. Esha must sit adjacent to Chetan. No two people can share a seat, and arrangements that differ only by rotation are considered the same. The lead also wants to know what happens if the condition that Farhan sits adjacent to Anita is dropped, because Farhan may join by video from another office and his physical seat could be left empty. The presentation coordinator needs a definite seating order and a short justification, not a guess.
1. Determine the seating arrangement around the table, numbering the seats 1 to 6 clockwise with Anita at seat 1. Justify each placement.
Read the worked answer
Place Anita at seat 1. Since Anita sits opposite Divya, Divya must be at seat 4.
Now test Chetan's possible seats for the condition 'Bala is second to the left of Chetan', i.e. two seats clockwise from Chetan. Chetan cannot be at 1 (Anita) or 4 (Divya).
- Chetan at 2 would put Bala at 4, but seat 4 is Divya-invalid.
- Chetan at 3 puts Bala at 5-valid so far.
- Chetan at 5 puts Bala at 1, but seat 1 is Anita-invalid.
- Chetan at 6 puts Bala at 2-valid so far.
For Chetan = 6, Bala = 2, the remaining seats 3 and 5 go to Esha and Farhan. Esha must be adjacent to Chetan, so Esha must be at 5. But seat 5 is adjacent to Divya at seat 4, violating the condition that Esha is not adjacent to Divya. So Chetan cannot be at 6.
Therefore Chetan is at 3 and Bala at 5. The remaining seats 2 and 6 go to Esha and Farhan. Esha must be adjacent to Chetan at 3; the adjacent seats are 2 and 4, and 4 is Divya, so Esha is at 2. Farhan is at 6.
Clockwise order: Anita (1), Esha (2), Chetan (3), Divya (4), Bala (5), Farhan (6).
2. The coordinator proposes dropping the condition 'Farhan sits adjacent to Anita' to simplify the problem. Does this change the solution? Explain whether the condition is necessary.
Read the worked answer
Dropping that condition does not change the solution, because it is redundant.
After placing Anita at 1, Divya at 4, Chetan at 3 and Bala at 5, only seats 2 and 6 remain for Esha and Farhan. Esha must sit adjacent to Chetan at 3. The neighbours of seat 3 are seats 2 and 4; seat 4 is occupied by Divya, so Esha must take seat 2. That forces Farhan into seat 6.
Seat 6 is adjacent to seat 1, where Anita sits. Therefore Farhan is adjacent to Anita automatically. The condition is true in every valid arrangement produced by the other constraints; removing it does not create a second arrangement. The coordinator should not treat it as an independent constraint when counting solutions.
3. Suppose the condition 'Esha sits adjacent to Chetan' is replaced by 'Esha sits adjacent to Bala'. Does a unique arrangement still exist? Give the resulting order.
Read the worked answer
Yes, a unique arrangement still exists.
Keep Anita at 1, Divya at 4, Chetan at 3 and Bala at 5. The remaining seats are 2 and 6 for Esha and Farhan. Esha must now be adjacent to Bala at seat 5. The neighbours of seat 5 are seats 4 and 6; seat 4 is Divya, so Esha must take seat 6. Farhan then takes seat 2.
Clockwise order: Anita (1), Farhan (2), Chetan (3), Divya (4), Bala (5), Esha (6).
Check the non-adjacency condition: Esha at 6 is adjacent to seats 5 and 1, not to Divya at 4, so the condition is satisfied. The replacement constraint still forces a single arrangement, though it is a different one from the original.