Seven Desks, One Claim of Uniqueness
A zonal office is seating seven LIC ADO trainees - A, B, C, D, E, F and G - at desks numbered 1 to 7 in a single row from left to right for a one-week induction. The seating must satisfy five conditions: A sits at one of the two end desks; B sits immediately to the right of F; C sits exactly three desks to the left of G; E sits at desk 4; and D does not sit adjacent to A. A clerk has prepared a chart showing one arrangement and claims the five conditions together fix a unique order. The zonal manager is not convinced and wants the full set of valid arrangements before the chart is printed. Desk 4 is also the only desk with a landline, so E's position cannot be shifted. Any change to the printed chart must be made before 5 p.m. today, and the manager wants a short written justification that can be filed with the induction records.
1. List every valid seating arrangement and test the clerk's claim that the conditions fix a unique order.
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Step 1 - place C and G. C must have exactly two desks between itself and G, so the C–G pair can be (1,4), (2,5), (3,6) or (4,7). Desk 4 is fixed for E, so C cannot be at 1 (that would put G at 4) and cannot be at 4. This leaves (C2, G5) and (C3, G6).
Step 2 - place F and B. In the first case the free desks are 1, 3, 6 and 7; the only adjacent pair available for F–B is 6–7, so F6 and B7. A must be at an end; desk 7 is taken, so A1 and D3. Arrangement: A C D E G F B.
In the second case the free desks are 1, 2, 5 and 7; the only adjacent pair is 1–2, so F1 and B2. A must be at an end; desk 1 is taken, so A7 and D5. Arrangement: F B C E D G A.
Conclusion. There are two valid arrangements, not one. The clerk's claim is false; both satisfy every condition, including D not being adjacent to A.
2. The clerk proposes adding one condition: 'F sits at an end.' Does this make the arrangement unique? If not, propose a different single condition that does.
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The added condition 'F sits at an end' does make the arrangement unique. In arrangement 1, F is at desk 6, which is not an end. In arrangement 2, F is at desk 1, which is an end. So only arrangement 2 survives: F, B, C, E, D, G, A.
If the manager wanted arrangement 1 instead, a cleaner single condition would be 'A sits at desk 1' or 'D sits at desk 3'. A condition such as 'C sits to the left of D' does not help, because C is to the left of D in both arrangements. The test is not whether a clue sounds restrictive, but whether it separates the two surviving cases. That is the same discipline needed in any data-arrangement question: generate the possible cases first, then ask what a new clue eliminates.
3. Is the condition 'D does not sit adjacent to A' redundant given the other four conditions? Justify your answer using the two arrangements.
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Given the other four conditions, the clue is redundant in this puzzle, though it is not redundant in general. Once the C–G pair and the F–B pair are placed, A is forced to the only remaining end desk and D takes the last free desk. In arrangement 1, A is at desk 1 and D at desk 3; in arrangement 2, A is at desk 7 and D at desk 5. Neither pair is adjacent, so the condition never does any work.
Removing it would not create a third valid arrangement, because the other clues already exhaust the end-desk options. A safe answer should say 'redundant under these conditions' rather than 'always redundant': if the C–G pair or the F–B pair could shift, D could become adjacent to A and the clue would matter.