Regional Rota Deduction
Four analysts - Ravi, Sok, Tamsin and Ugo - must each be assigned to exactly one of four regions: Central, Delta, Hills and Port. Each region is covered by exactly one analyst. The assignment must satisfy five constraints simultaneously:
- Ravi is not assigned to Hills.
- Sok is assigned to Central or Delta.
- Ugo is assigned to Port only if Tamsin is assigned to Hills.
- Tamsin is not assigned to Delta.
- Ugo is not assigned to Hills.
The rota is published weekly, and moving an analyst after publication requires a two-day handover, so the operations lead wants the deduction settled on paper before anything is circulated. She has asked you not to guess: every conclusion must follow from the constraints alone, not from a single workable example.
1. Which analyst must be assigned to Hills? Justify the answer from the constraints rather than by testing arrangements.
Read the worked answer
Answer: Tamsin must be assigned to Hills.
Deduction. Hills must be covered by exactly one of the four analysts. Eliminate the others:
- Ravi cannot be in Hills - constraint 1 says so directly.
- Sok cannot be in Hills - constraint 2 restricts Sok to Central or Delta.
- Ugo cannot be in Hills - constraint 5 says so directly.
That leaves Tamsin, and constraint 4 (Tamsin is not in Delta) does not block Hills.
Why this is a proof and not an example. The argument does not construct a full arrangement and check it; it uses the covering requirement (every region filled exactly once) plus three exclusion rules to force the assignment. Any valid arrangement must therefore have Tamsin in Hills, whatever else is true.
Useful side-effect. Once Tamsin is fixed in Hills, constraint 3 ('Ugo is in Port only if Tamsin is in Hills') is automatically satisfied, so Ugo may take Port, Central or Delta - the conditional no longer restricts anything. Recognising that removes a constraint from the remaining search and is the quickest route to the rest of the rota.
2. If Ravi is assigned to Port, what follows? State what is fixed and what remains open.
Read the worked answer
Fixed.
- Ravi is in Port (given).
- Tamsin is in Hills (from the earlier deduction: Hills can only be Tamsin).
What remains. The two remaining regions, Central and Delta, must go to Sok and Ugo in some order. Both orders satisfy every constraint:
- Order 1: Sok–Central, Ugo–Delta. Sok is in Central (constraint 2 satisfied); Ugo is not in Port, so constraint 3 is not triggered; Ugo is not in Hills.
- Order 2: Sok–Delta, Ugo–Central. Sok is in Delta (constraint 2 satisfied); same reasoning on constraints 3 and 5.
What must therefore be true. Ugo is not in Port, Sok is not in Hills, and Tamsin is in Hills. What is not determined: whether Sok takes Central or Delta.
Why the conditional in constraint 3 does not help here. Constraint 3 reads 'Ugo is in Port only if Tamsin is in Hills'. It is a one-way condition: it forbids Ugo in Port with Tamsin elsewhere, but it does not require Ugo to be in Port when Tamsin is in Hills. Treating it as a two-way condition is the most common error on this puzzle; it would wrongly force Ugo into Port and produce a single arrangement where two are permitted.
3. A colleague circulates this draft: Sok–Delta, Ravi–Central, Tamsin–Hills, Ugo–Port. Verify it against all five constraints and state whether it can be published.
Read the worked answer
Check each constraint in turn.
- Ravi is not in Hills - Ravi is in Central. Satisfied.
- Sok is in Central or Delta - Sok is in Delta. Satisfied.
- Ugo is in Port only if Tamsin is in Hills - Ugo is in Port, so the condition is triggered and Tamsin must be in Hills. Tamsin is in Hills. Satisfied.
- Tamsin is not in Delta - Tamsin is in Hills. Satisfied.
- Ugo is not in Hills - Ugo is in Port. Satisfied.
Coverage check. Central, Delta, Hills and Port each appear exactly once across the four analysts. No region is duplicated and none is empty.
Verdict: valid and publishable. This is one of the permitted arrangements, not the only one: Sok–Central with Ugo–Delta would also work, since Ugo is not then in Port and constraint 3 is never triggered.
The distinction that matters. A draft that satisfies every constraint proves the rota is possible; it does not prove the rota is the only option or the best one. If the operations lead asks 'is this the only answer?', the correct response is no - and the valid alternatives should be listed before publication, so that a later change is not mistaken for an error.