Fixing the Sentry Order from Five Constraints
Four soldiers - A, B, C and D - are posted for the night to four sentry posts, P1 to P4, on a perimeter road running west to east. The orderly officer has five written constraints in the duty notebook, and he wants the post order fixed before he hands over at 2200: (1) A is not to be at P1. (2) B must be immediately to the right of C. (3) D must be at one of the two end posts. (4) C must not be at P3. (5) A must not be posted next to D. The relief arrives at 0200, and the officer also wants to know how far the plan depends on each rule.
1. Determine the single post order that satisfies all five constraints.
Read the worked answer
Constraint (2) fixes a block (C, B) occupying consecutive posts. It can only sit at (P1,P2), (P2,P3) or (P3,P4). Test the three positions against the rest.
Case 1 - (C,B) at P1,P2. Remaining posts P3 and P4 go to A and D. Constraint (3) forces D to an end, so D = P4 and A = P3. But then A at P3 sits next to D at P4, breaching (5). Rejected.
Case 2 - (C,B) at P2,P3. Remaining posts P1 and P4. Constraint (1) bars A from P1, so A = P4 and D = P1. Check (3): D at P1 is an end ✓. Check (5): P4 and P1 are not adjacent ✓. Accepted.
Case 3 - (C,B) at P3,P4. Rejected at once by (4), since C would be at P3.
Order: D – C – B – A across P1 to P4. Note that (5) does the decisive work - it is the rule that kills the otherwise attractive Case 1.
2. The adjutant claims that rule (4), 'C must not be at P3', can be struck out without changing the answer. Is he right? Show your working.
Read the worked answer
He is right. Re-run the cases with (4) deleted.
- (C,B) at P1,P2 - D must still take an end, so D = P4, A = P3, and A is adjacent to D, breaching (5). Still rejected.
- (C,B) at P2,P3 - A ≠ P1 forces A = P4, D = P1. Valid, as before.
- (C,B) at P3,P4 - now permitted by default, but remaining posts are P1 and P2 for A and D. D must be at an end, so D = P1 and A = P2. A at P2 is adjacent to D at P1, breaching (5). Rejected.
So (4) removes a case that constraint (5) already removes independently: it is redundant, and the answer stays D – C – B – A.
Judgement for the officer: redundancy is not a reason to delete the rule from the notebook. A redundant constraint still documents intent and will become load-bearing the moment any other rule is amended. Keep it, but know that (5) is the rule actually doing the work - if (5) is ever relaxed, (4) alone will not save the plan.
3. If rule (5) is replaced by 'A must be posted immediately to the left of D', how many valid orders remain?
Read the worked answer
Treat (A, D) as a second block and intersect it with the (C, B) block, under (1), (3) and (4).
Possible (A, D) positions: (P1,P2), (P2,P3), (P3,P4). Rule (1) eliminates (P1,P2).
- (A,D) at P2,P3 - the remaining posts P1 and P4 cannot hold the adjacent block (C,B). Impossible.
- (A,D) at P3,P4 - (C,B) must take P1,P2, which respects (4) since C lands at P1. Order: C – B – A – D. Check (3): D at P4 is an end ✓.
Any other arrangement fails because the two blocks would overlap. Exactly one valid order remains, so the amended rule set is still uniquely solvable.
Staff point: a unique answer is not the same as a robust plan. Both rule sets yield one order, but the orders differ, so the officer must confirm which version is currently in force before he hands over - otherwise the 0200 relief can be briefed from the wrong notebook page.