Seven Posts and a Sergeant's Written Order
A drill sergeant must place seven constables - A, B, C, D, E, F and G - at seven numbered posts, 1 to 7, in a single line facing the saluting dais. Post 4 is the band leader's mark and must go to the tallest constable, who is D (178 cm). The written order adds four conditions: C must stand to the left of B, with B immediately next to C on C's right; A must not occupy either end post; E and F must have exactly two constables standing between them; and G must not stand next to A. A havildar points out that the order says nothing about height for any post except 4, and asks whether the line-up is fixed or whether several arrangements satisfy it. The sergeant also wants to know which single extra condition would settle the matter, and whether any of the four existing conditions is doing no work at all. The parade falls in at 06:00 and the sergeant has twenty minutes to sign the final chart.
1. How many arrangements satisfy all the conditions? Show the deduction rather than guessing.
Read the worked answer
Fix D at post 4. E and F must be three posts apart, so the possible pairs are (1,4), (2,5), (3,6) and (4,7). Since post 4 is taken, only (2,5) and (3,6) survive. If E and F hold 2 and 5, the adjacent pair C–B cannot touch 2, 4 or 5, so it must be (6,7): C at 6, B at 7. Posts 1 and 3 remain for A and G, and A cannot sit at an end, so A=3 and G=1; G is not next to A. If E and F hold 3 and 6, C–B must be (1,2), leaving 5 and 7 for A and G; A cannot take 7, so A=5 and G=7. Each branch still allows E and F to swap, so there are two arrangements per branch and four in all. The line-up is not fixed: four arrangements satisfy every condition.
2. The sergeant adds one condition - B must occupy an odd-numbered post. Who now stands at post 3, and is the answer unique?
Read the worked answer
The new condition eliminates the second branch outright, because there B sits at post 2, an even number. Only the first branch survives: G=1, E/F=2, A=3, D=4, F/E=5, C=6, B=7. Post 3 is therefore A. The answer is unique even though the full line-up is not, because the only remaining freedom is the E/F swap at posts 2 and 5, which does not touch post 3. This is worth stating to the sergeant: uniqueness of one post does not mean the chart is settled. Had he instead required B at an even post, the surviving branch would put A at post 5, and signing the old chart would have placed the wrong man at post 3.
3. Is any of the four original conditions redundant? Explain how you would test that.
Read the worked answer
Test a condition by removing it and re-solving; if the solution set does not change, the condition does no work. Doing that here, the conditions fixing D=4, the C–B adjacency, the E–F separation and A's exclusion from the ends already force the four arrangements. In every one of them G lands at an end (1 or 7) while A lands at 3 or 5, so G is never adjacent to A. The clause 'G not next to A' therefore eliminates no arrangement and is redundant. It should stay in the order only as a check, not as a constraint that the sergeant believes is shaping the line-up. Redundancy is not harmless: it misleads the person drafting the order into thinking a degree of freedom has been closed when it has not.