Seating Six Probationary Officers When One Condition Makes the Puzzle Unsolvable
A bank must place six probationary officers - A, B, C, D, E and F - across three floors (floor 1 lowest, floor 3 highest) for a two-week training rotation, with exactly two officers on each floor. The coordinator gives five conditions: (i) A and B must be on different floors; (ii) C must be on the floor immediately above D's floor; (iii) E must not be on floor 1; (iv) F must be on the same floor as A; (v) B must be on a higher floor than C. At the last minute the coordinator adds that both officers on floor 3 must come from the group {A, B, E}. A colleague has already circulated a tentative arrangement: floor 1 - C and F; floor 2 - D and E; floor 3 - A and B. Nobody has checked it against all five conditions. You have ten minutes before the schedule is printed and pinned on the notice board.
1. Test the circulated arrangement against each of the five conditions and report exactly which ones it fails.
Read the worked answer
Place the arrangement and test each rule mechanically:
- Floor 1: C, F. Floor 2: D, E. Floor 3: A, B.
- (i) A and B on different floors - A and B are both on floor 3. Fails.
- (ii) C immediately above D - C is on floor 1 and D on floor 2, so C is below D, not above. Fails.
- (iii) E not on floor 1 - E is on floor 2. Satisfied.
- (iv) F on the same floor as A - F is on floor 1, A on floor 3. Fails.
- (v) B higher than C - B is on 3, C on 1. Satisfied.
The arrangement fails three of the five conditions, so it cannot be printed with a correction note; it has to be rebuilt. Two of the failures - (i) and (iv) - are structural, not clerical: they involve pairs who must be separated or kept together, which means the whole layout is wrong rather than one name being in the wrong slot. Report that clearly before anyone starts moving single names around.
2. The coordinator's new condition requires both floor-3 officers to be from {A, B, E}. Determine whether any arrangement can satisfy all five conditions plus this one.
Read the worked answer
Enumerate the floor-3 pairings from {A, B, E} - there are only three possibilities:
- {A, B} - directly violates (i), which requires A and B on different floors.
- {A, E} - condition (iv) forces F onto A's floor, so floor 3 would hold A, E and F. That breaches the two-per-floor limit.
- {B, E} - floors 1 and 2 must then hold A, C, D and F. Condition (iv) puts A and F together, and since each floor holds exactly two, A and F occupy one entire floor. C and D are left as the pair on the other floor, but (ii) requires C to be immediately above D - impossible when they share a floor.
Every branch fails, so no arrangement exists. The added condition is inconsistent with (i), (ii) and (iv) jointly, not merely awkward. The correct output is a statement that the constraints are mutually contradictory, with the three-case reasoning, rather than a plausible-looking chart produced under time pressure. Printing any chart here guarantees that at least one stated condition is broken and the error will be discovered by the officers themselves.
3. You cannot change the floor-3 condition or the ten-minute deadline. Which single existing condition would you ask the coordinator to relax, and how would you verify the resulting schedule?
Read the worked answer
From the three-case analysis, the {B, E} branch is the only one that survives to the stage of placing A, C, D and F, and it collapses solely because (iv) packs A and F into one floor. So (iv) is the cheapest single condition to drop.
With (iv) relaxed, try: floor 1 - D and A; floor 2 - C and F; floor 3 - B and E. Check: (i) A on 1, B on 3 - different floors, satisfied. (ii) C on 2 sits immediately above D on 1, satisfied. (iii) E on 3, not floor 1, satisfied. (v) B on 3 is higher than C on 2, satisfied. Floor-3 pair {B, E} is within {A, B, E}. All conditions hold.
Before printing, confirm with the coordinator that (iv) is a preference - keeping a mentor and mentee together - rather than a safety or supervision requirement, because a rule with an operational reason should be the one that stays while a different rule gives way. Say plainly that the deadline did not cause the conflict; the added condition did, and one rule must move.