Scheduling Four Case Reviews Under Conflicting Constraints
A compliance team must review four case files - W, X, Y and Z - one per day from Monday to Thursday. Each file is reviewed exactly once. The standing rules are:
- X is reviewed before Z.
- Y is not reviewed on Monday.
- W is reviewed on the day immediately after Y.
A supervisor claims that rule set already fixes who goes first, and asks the team to treat "X is reviewed on Monday" as settled. A new analyst wants to add one further condition so that the whole week is determined.
1. How many schedules satisfy the three rules, and what are they?
Read the worked answer
Rules 2 and 3 together work as a block. W comes immediately after Y, so (Y, W) occupies two consecutive days. Y cannot be Monday, so the block cannot start on Monday.
Possible block positions: (Tue, Wed) or (Wed, Thu).
Case A - block on Tue/Wed: Y = Tue, W = Wed. Days Mon and Thu are left for X and Z. Rule 1 wants X before Z, so X = Mon, Z = Thu. → X, Y, W, Z
Case B - block on Wed/Thu: Y = Wed, W = Thu. Days Mon and Tue are left for X and Z, and again X before Z. → X, Z, Y, W
Exactly two schedules satisfy the rules. Any answer that fixes the week completely after three rules has over-constrained the puzzle; the block rule constrains relative position, not absolute position.
2. Is the supervisor's claim, that X is reviewed on Monday, guaranteed? Prove it without relying on the two listed schedules.
Read the worked answer
Yes - it is entailed, and the direct proof is short.
Y can only be Tuesday or Wednesday (rule 2 excludes Monday; rule 3 needs a following day, so Thursday is excluded too).
- If Y is Tuesday, W is Wednesday, so Monday must hold X or Z. Monday cannot hold Z, because rule 1 requires X before Z, and nothing precedes Monday. Therefore Monday = X.
- If Y is Wednesday, W is Thursday, so Monday and Tuesday hold X and Z in some order, with X first. Therefore Monday = X.
There is no third option, so X is Monday in every valid schedule. The supervisor is right, though not for the reason she gave: it is a consequence of the block rule and the "not Monday" rule squeezing Y into the middle of the week.
Caution for the team: the claim is true only while all three rules stand. If rule 2 were relaxed, (Y, W) could sit on Mon/Tue and X would move to Wednesday or Thursday, so the claim would collapse.
3. Which single additional condition would make the schedule unique? Give at least two that work and check each.
Read the worked answer
We need a condition true in exactly one of {X, Y, W, Z} and {X, Z, Y, W}.
Works - "Z is reviewed on Tuesday." Z = Tue forces Case B, since in Case A Z is Thursday. Schedule: X, Z, Y, W. Unique. Also note Z on Tuesday already forces X to Monday (rule 1), so the condition is doing real work.
Works - "W is reviewed on Thursday." W = Thu puts the block at Wed/Thu (Case B only). Schedule: X, Z, Y, W. Unique.
Works - "Y is reviewed on Wednesday." Directly selects Case B.
Does not work - "X is reviewed on Monday." Already true in both cases, so it discriminates nothing.
Does not work - "Z is not reviewed on Monday." Trivially true everywhere.
Test to apply to any proposal: build the two candidate schedules first, then check whether the new condition is true in one and false in the other. A condition that is already implied cannot break the tie - this is the commonest error in "add one rule" questions.