The missing file: reconstructing a four-way audit trail
Four correction files - Pension, Land record, Ration card and Caste certificate - were cleared on the same day by four clerks, W, X, Y and Z, one file each, in four consecutive hourly slots: 10:00, 11:00, 12:00 and 13:00. The audit note records six statements, all true on the day: (i) X worked immediately before the ration card file. (ii) The pension file was cleared at 10:00. (iii) Y cleared the caste certificate. (iv) Z was not the first clerk of the day and did not clear the ration card file. (v) W cleared either the land record or the ration card file. (vi) The land record file was cleared before the caste certificate. The file has since gone missing and the audit section needs the slot-clerk-file mapping reconstructed. A junior has already claimed that statement (v) can be struck out because it adds nothing, and a loose note in the file says the caste certificate was cleared at 12:00.
1. Reconstruct the full mapping, showing the elimination of each rejected possibility.
Read the worked answer
Pension is at 10:00 (ii), so the ration card must be at 11:00, 12:00 or 13:00.
- Ration at 12:00 → X at 11:00 (i). Slots 10:00, 12:00 and 13:00 remain for W, Y, Z, and the ration file at 12:00 cannot be Y's (iii) or Z's (iv), so W is at 12:00. That leaves Y and Z for 10:00 and 13:00; Y cannot take 10:00 (that is pension, and Y has the caste file) and Z cannot take 10:00 (iv). Rejected.
- Ration at 13:00 → X at 12:00 (i). No clerk can take 10:00: Y is excluded by (iii), Z by (iv), and W, whose file is land or the ration card (v), cannot hold pension, which is what 10:00 is. Rejected.
- So ration is at 11:00 and X at 10:00 with pension. Slots 12:00 and 13:00 go to Y and Z. Y cannot be at 11:00 (ration, and Y has caste) and Z cannot (iv), so W is at 11:00 with the ration card. (vi) then forces land before caste: Z – land record at 12:00, Y – caste certificate at 13:00.
Mapping: 10:00 X pension; 11:00 W ration card; 12:00 Z land record; 13:00 Y caste certificate. It satisfies all six statements.
2. Test the junior's claim that statement (v) is redundant.
Read the worked answer
The claim is correct. Re-read the reconstruction above: (v) is never used.
The ration card cannot be at 12:00, because that forces W into 12:00 and leaves 10:00 to Y or Z, each barred by (iii) or (iv). It cannot be at 13:00, because then X occupies 12:00 and no clerk can hold pension at 10:00 - Y has caste, Z is barred by (iv), and the only other candidate is W, who by (v) would have land or the ration card and so could not hold pension. (That last step does use (v); without it, test ration at 13:00 again: W at 10:00 pension, X at 12:00, and slot 13:00 must be filled by Y or Z with the ration card - Y has caste and Z is barred by (iv). Still impossible.)
So the ration card is at 11:00 and X at 10:00 with pension. Of the three remaining clerks, Y cannot take 11:00 (caste) and Z cannot (iv), so W takes 11:00, and therefore the ration card. (v) is true as a consequence of (i)-(iv), not as an independent constraint.
Conclusion: (v) may be struck out without changing the solution. It remains useful as a consistency check, which is why it should be retained in the note rather than deleted.
3. The loose note says the caste certificate was cleared at 12:00. Reconcile this with the six statements.
Read the worked answer
If caste is at 12:00, then Y is at 12:00 (iii). The ration card cannot be at 12:00 (Y is there) and cannot be at 13:00, because (i) would put X at 12:00, which Y already occupies. So ration is at 11:00, X is at 10:00 with pension, and W is at 11:00 with the ration card. Only slot 13:00 and the land record remain, so Z is at 13:00 with the land file.
Check all six: (i) X at 10:00 immediately before ration at 11:00 - true. (ii) pension at 10:00 - true. (iii) Y has caste - true. (iv) Z is neither first nor on ration - true. (v) W has the ration card - true. (vi) land before caste - false, because land is at 13:00 and caste at 12:00.
So the loose note is consistent with five of the six statements and inconsistent with exactly one: (vi) is the false statement. The audit should record (vi) as unverified rather than treat the whole note as unreliable, and the mapping to report is 10:00 X pension, 11:00 W ration, 12:00 Y caste, 13:00 Z land.