Five Credit Files and Four Clues: How Far Does the Arrangement Actually Constrain?
The audit team of a district branch has to rearrange five pending credit files - P, Q, R, S and T - on a shelf that has exactly five numbered slots, 1 (leftmost) to 5 (rightmost). The record keeper has left four written clues, and the audit has to be closed within the next thirty minutes:
(i) S must occupy slot 5. (ii) Q must be placed immediately to the right of P. (iii) T must be placed somewhere to the left of R. (iv) R must be placed somewhere to the left of P.
The branch head is unsure whether these four clues are enough to fix every file, and whether the answer changes if one of the clues is relaxed. She also wants to know how much the clues really constrain the arrangement, because a vendor will print a new shelf layout only if the position of every file is certain.
1. Fix the position of each of the five files. Show how the clues combine and why no other arrangement survives.
Read the worked answer
Step 1 - build the block. Clue (ii) fixes the two-slot block P–Q, with P immediately to the left of Q.
Step 2 - chain the clues. Clues (iii) and (iv) give T < R < P. Combining with the block gives the chain T < R < P–Q, which needs four consecutive slots.
Step 3 - place the chain. Clue (i) locks slot 5 to S, so slots 1–4 must accommodate T, R, P and Q in exactly that order:
| Slot | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| File | T | R | P | Q | S |
Why the alternative fails. The chain of four cannot start at slot 2: that would place P at 4 and Q at 5, but slot 5 is already S. So the chain must start at slot 1. Verification: S is at 5; Q is immediately right of P (3→4); T (1) is left of R (2); R (2) is left of P (3). All four clues hold, and the arrangement is unique.
2. A colleague suggests replacing clue (ii) with the weaker statement 'Q must be placed somewhere to the right of P'. Test the claim that the arrangement stays uniquely determined.
Read the worked answer
Weaken the clue and re-derive. Now we only know T < R < P, Q > P, and S = 5.
The block is gone, so test each possible position of P over slots 1–4:
- P = 2: T and R must both lie left of P, but only slot 1 is available for two files - impossible.
- P = 3: Q must be to the right, and slot 5 is taken by S, so Q = 4. Slots 1 and 2 remain for T and R, and T < R forces T = 1, R = 2. One arrangement: T, R, P, Q, S.
- P = 4: Q must be to the right, and the only such slot is 5, which S occupies - impossible.
Only one arrangement survives, and it is the same one. The claim is correct: for this set of clues, 'Q immediately right of P' adds nothing, because the end slot being reserved by S leaves Q only one legal place. This is worth noting in the audit file - a clue that looks structural is actually redundant, which means the layout is less constrained than it appears and would break if any one of the other three clues were changed.
3. If clue (i) is withdrawn, how many valid arrangements remain, and does your first answer still hold?
Read the worked answer
Without S = 5, arrange T, R, P, Q, S under: Q immediately right of P, T < R < P. Count by P's position (P must have at least two files to its left and one slot to its right for Q):
- P = 3, Q = 4: slots 1–2 hold T and R with T < R, so T = 1, R = 2. S takes slot 5. 1 arrangement.
- P = 4, Q = 5: slots 1–3 hold T, R and S with T < R. Three ways: (S,T,R), (T,S,R), (T,R,S). 3 arrangements.
- P = 2 or P = 5: impossible (two files cannot fit to the left of slot 2; nothing fits right of 5).
Total: 4 arrangements. The original order T, R, P, Q, S is one of them, so every clue in the first answer remains true - but it is no longer the only solution. Practical consequence: the printed layout cannot be certified from clues (ii)–(iv) alone. Either clue (i) stays, or the vendor must be told that four layouts are consistent with the record and that the shelf labelling, not the clues, will have to decide.