Ranking Five Candidates from Four Clues
Five candidates - A, B, C, D and E - appear for a departmental entrance test and obtain distinct scores. A notice board lists four clues about their relative performance, but the clerk who wrote them is known to be careless, so each clue must be verified rather than assumed:
- C scored higher than exactly two of the other candidates.
- D scored higher than A but lower than B.
- E did not score the lowest.
- B did not score the highest. A student argues that because the clerk is unreliable, no definite ranking can be derived. Another claims the ranking is fully determined. Both agree that 'higher than exactly two' means C is third, with two candidates above and two below. The candidates want to know the actual order, whether every clue is doing real work, and what happens if the fourth clue is withdrawn.
1. Derive the full ranking from the four clues, showing where each clue is used.
Read the worked answer
Clue 1: C is 3rd. Clue 2: B > D > A. Clue 4: B is not 1st.
Places 4 and 5 must be filled by two of {A, D, E}, but E must be above C (clue 3) and C is 3rd, so E is 1st or 2nd. Hence places 4 and 5 go to A and D, and since D > A, D = 4th, A = 5th.
That leaves places 1 and 2 for B and E. Clue 4 rules out B = 1st, so B = 2nd, E = 1st.
Order: E > B > C > D > A.
Check against clue 3: E is 1st, not lowest - satisfied. All four clues hold, so the careless clerk happened to write a consistent set, and the ranking is unique.
2. Is clue 3 doing any work? Justify your answer by deriving the order without it.
Read the worked answer
Drop clue 3 and re-derive. Clue 1 gives C = 3rd. Clue 2 gives B > D > A, so D and A are both below B. Clue 4 says B is not 1st.
Places 1 and 2 must therefore be occupied by B and E. B cannot be 1st (clue 4), so B = 2nd and E = 1st. Places 4 and 5 go to D and A, with D > A, so D = 4th and A = 5th.
Same order: E > B > C > D > A.
Conclusion: clue 3 is redundant - it is implied by clues 1, 2 and 4 rather than needed to pin the order down. A redundant clue is not a false clue, but it is worthless as evidence and dangerous in a logical-reasoning question, because it invites solvers to 'use' information that adds no constraint. The exam-relevant habit is to test necessity clue by clue, not just consistency of the final arrangement.
3. If clue 4 is withdrawn instead, how many distinct rankings satisfy the remaining clues, and what are they?
Read the worked answer
With clues 1, 2 and 3 only:
- C = 3rd (clue 1).
- B > D > A (clue 2).
- E is not last (clue 3).
Places 4 and 5 must be D and A in that order, since D > A and both are below B. So D = 4th, A = 5th, and A is last - which satisfies clue 3 automatically (E is not last).
Places 1 and 2 are now free for B and E, with no clue 4 to forbid B = 1st.
Two rankings satisfy the clues:
- B > E > C > D > A
- E > B > C > D > A
So withdrawing clue 4 costs exactly one bit of certainty: the top two can swap. Clue 4 is therefore the load-bearing clue, while clue 3 is decorative - the opposite of what their ordering on the notice board suggests.