Locked Rack: Reconstructing a Controlled-Drug Shelf Order
A 200-bed district hospital keeps five controlled drugs on a locked vertical rack with five numbered shelves, shelf 1 at the top and shelf 5 at the bottom: Fentanyl, Naloxone, Tramadol, Morphine and Pethidine. The narcotics register records four standing rules, and the outgoing pharmacist must sign the rack count before every shift change:
(i) Naloxone sits immediately below Fentanyl. (ii) Morphine is on neither the top nor the bottom shelf. (iii) Tramadol is somewhere above Pethidine, but not immediately above it. (iv) Fentanyl occupies an odd-numbered shelf above shelf 3.
A newly joined pharmacist must reconstruct the arrangement from these rules alone, because the shelf labels have faded. Later the same week a second pharmacist argues that rule (iv) can be relaxed to "Fentanyl is on an even-numbered shelf" without affecting the count, and a third asks whether any one rule could be deleted altogether.
1. Reconstruct the shelf arrangement from rules (i)–(iv), showing each deduction, and state which drug occupies shelf 3.
Read the worked answer
Step 1 - Place Fentanyl. Rule (iv) requires an odd-numbered shelf above shelf 3. The odd shelves are 1, 3 and 5; only shelf 1 is above 3. So Fentanyl = shelf 1.
Step 2 - Place Naloxone. Rule (i) puts it immediately below Fentanyl, so Naloxone = shelf 2.
Step 3 - Allocate the remaining three shelves (3, 4, 5) to Morphine, Tramadol and Pethidine. Rule (iii) says Tramadol is above Pethidine but not directly above. Test the possible Tramadol/Pethidine pairs from {3,4,5}:
- (3,4): directly above - barred.
- (4,5): directly above - barred.
- (3,5): above, with one shelf between - allowed.
So Tramadol = shelf 3, Pethidine = shelf 5, and Morphine takes the only remaining shelf, shelf 4.
Step 4 - Verify rule (ii). Morphine on shelf 4 is neither top nor bottom. Consistent.
Final order: 1 Fentanyl, 2 Naloxone, 3 Tramadol, 4 Morphine, 5 Pethidine. The drug on shelf 3 is Tramadol.
2. Evaluate the second pharmacist's claim that replacing rule (iv) with "Fentanyl is on an even-numbered shelf" leaves the arrangement uniquely determined. Construct all valid arrangements to support your answer.
Read the worked answer
The claim is false. With Fentanyl on an even shelf it may sit on shelf 2 or shelf 4, and both branches survive the remaining rules:
Branch A - Fentanyl on shelf 2. Naloxone must be immediately below, on shelf 3. Shelves 1, 4 and 5 are left for Morphine, Tramadol and Pethidine. Rule (ii) bars Morphine from 1 and 5, forcing Morphine to shelf 4. Tramadol must be above Pethidine and not adjacent, and with shelves 1 and 5 free the pair (1,5) satisfies this. Arrangement: Tramadol 1, Fentanyl 2, Naloxone 3, Morphine 4, Pethidine 5.
Branch B - Fentanyl on shelf 4. Naloxone goes to shelf 5. Shelves 1, 2 and 3 are left. Among these, the only above-but-not-adjacent pair is (1,3), so Tramadol = 1 and Pethidine = 3, leaving Morphine on shelf 2 (permitted, since it is neither top nor bottom). Arrangement: Tramadol 1, Morphine 2, Pethidine 3, Fentanyl 4, Naloxone 5.
Two distinct arrangements therefore satisfy the relaxed rule, so the count is no longer unique. The precise wording "odd-numbered shelf above shelf 3" is what pins Fentanyl to shelf 1 and makes the record auditable.
3. Which single rule among (i)–(iv) is redundant, in the sense that deleting it still leaves a unique arrangement? Justify your answer, and show why deleting a different rule would not.
Read the worked answer
Rule (ii) is redundant. Keep (i), (iii) and (iv):
- (iv) forces Fentanyl to shelf 1 (the only odd shelf above 3).
- (i) then forces Naloxone to shelf 2.
- Shelves 3, 4 and 5 hold Morphine, Tramadol and Pethidine, and (iii) admits only the pair (3,5) for Tramadol/Pethidine - the pairs (3,4) and (4,5) are adjacent and barred.
- Morphine therefore falls on shelf 4 by elimination, which happens to satisfy (ii) automatically.
So (ii) is a check, not a constraint: it can be removed without creating a second solution.
Why not another rule? Delete (iii) and the arrangement collapses into several possibilities: with Fentanyl on 1 and Naloxone on 2, Tramadol and Pethidine may occupy (3,4), (4,5) or (3,5) in either order, and Morphine fills the leftover shelf where rule (ii) allows. Delete (iv) and Fentanyl may sit on shelves 1 or 3, multiplying the branches. Deleting (i) similarly frees Naloxone. Only (ii) can go while the answer stays unique.