Settling the Order of Five Disposal Files
A record room has five numbered racks, 1 (topmost) to 5 (bottom). Five disposal files - P, Q, R, S and T - must each occupy one rack before Monday's audit. The caretaker has left a note of the standing instructions:
- R must be on the rack immediately below Q.
- T must not be on the topmost or the bottommost rack.
- P must be somewhere above S.
- S must not be on a rack adjacent to R.
- Q must not be on rack 1.
A new assistant arranges the files as Q, R, P, T, S from top to bottom and claims the note permits it. The audit officer says the arrangement is wrong, and further claims that one of the five instructions is redundant because the other four already fix a single order. The section head wants the arrangement settled, the officer's claim tested, and a note prepared on what happens if instruction 4 is withdrawn.
1. Determine the arrangement consistent with all five instructions, showing how the alternatives are eliminated.
Read the worked answer
Treat QR as a single block and try each of its four possible positions.
- QR on (1,2): Q=1, R=2. Remaining racks 3,4,5 for P,S,T. Instruction 2 forces T to rack 4; then P=3, S=5, and S is not adjacent to R. Order: Q R P T S - but Q on rack 1 violates instruction 5. Rejected.
- QR on (2,3): remaining 1,4,5. T must be on 4; P above S forces P=1, S=5. S is not adjacent to R (rack 3 vs 5). Order: P Q R T S - satisfies all five.
- QR on (3,4): remaining 1,2,5. T must be on 2; P above S forces P=1, S=5; but S on 5 is adjacent to R on 4. Rejected.
- QR on (4,5): remaining 1,2,3. T must be on 2; P above S forces P=1, S=3; S on 3 is adjacent to R on 4. Rejected.
Only one order survives: P, Q, R, T, S from top to bottom. The new assistant's arrangement fails instruction 5, and it also violates nothing else only by luck - it is one of the two orders that survive when instruction 5 is ignored.
2. Test the audit officer's claim that one of the five instructions is redundant.
Read the worked answer
The claim is false: every one of the five is load-bearing. With the true answer P Q R T S, remove one instruction at a time and exhibit a second valid order.
- Without 1: P R Q T S works (R is not immediately below Q, T is on 4, P is above S, S is not adjacent to R on rack 2, Q is not on rack 1).
- Without 2: Q R P S T works (T sits on the bottom rack, which is now allowed).
- Without 3: S T Q R P works (S on 1 is not adjacent to R on 4, T on 2, Q not on rack 1).
- Without 4: P T Q R S works (S on 5 is adjacent to R on 4, which is now permitted).
- Without 5: Q R P T S works - the assistant's own arrangement.
So each instruction eliminates at least one order that the remaining four would allow, and the officer's remark should be rejected in writing. The correct finding is that the instruction set is minimal for a unique solution, which is a stronger and more useful statement for the audit note.
3. If instruction 4 is withdrawn, how many arrangements satisfy the remaining four instructions, and which are they?
Read the worked answer
Three arrangements survive.
- QR on (1,2): Q=1 violates instruction 5. Excluded.
- QR on (2,3): T=4, P=1, S=5 → P Q R T S.
- QR on (3,4): T=2, P=1, S=5 → P T Q R S. This is now admissible because S on 5, adjacent to R on 4, is no longer barred.
- QR on (4,5): T=2, P=1, S=3 → P T S Q R. Also newly admissible, again because of the withdrawn adjacency bar.
So the answer is three, and only one of them (P Q R T S) was admissible under all five instructions. Two conclusions follow for the section head: the adjacency instruction was doing real work rather than merely restating the others, and any relaxation of it must be accompanied by a fresh tie-breaking rule - for example, a rule on which file must sit directly above the other - if a single auditable order is still required.