Five Cadets, Five Simulator Slots
A squadron must fill five simulator slots - 0800, 0900, 1000, 1100 and 1200 - with five cadets: Ali, Bilal, Cheema, Danish and Ejaz. One cadet per slot, each cadet flies once. Number the slots 1 to 5 in that order. The scheduling clerk's file records only these conditions:
- Cheema flies immediately before Bilal.
- Danish is given a slot with an even number.
- Ali is neither first nor last.
- Ejaz flies earlier than Ali.
- Cheema is not in the 0800 slot.
The clerk must publish the timetable within the hour. The flight commander wants to know not only the timetable but which entries are forced by the conditions and which are merely convenient, and whether any of the five conditions is doing no work at all.
1. Derive the complete timetable. Show why the apparently attractive placements fail, not only the one that succeeds.
Read the worked answer
Method: test the adjacent pair first, since it is the most restrictive clue. Cheema-Bilal can occupy (1,2), (2,3), (3,4) or (4,5). Rule 5 removes (1,2).
Try (2,3): Cheema = 2, Bilal = 3. Danish must be 4 (2 is taken). Slots 1 and 5 remain for Ali and Ejaz, but rule 3 forbids Ali in either - fails.
Try (3,4): Cheema = 3, Bilal = 4. Danish must be 2 (4 is taken). Slots 1 and 5 remain for Ali and Ejaz, and Ali again has nowhere legal - fails.
Try (4,5): Cheema = 4, Bilal = 5. Danish must be 2 (4 is taken). Slots 1 and 3 remain. Ali cannot take 1, so Ali = 3 and Ejaz = 1. Rule 4 (Ejaz before Ali) is satisfied - works.
Timetable: 0800 Ejaz, 0900 Danish, 1000 vacant-nothing - corrected: 1000 is the remaining cadet position held by nobody in the pair, so the full order is
- 0800 Ejaz
- 0900 Danish
- 1000 Cheema
- 1100 Ali
- 1200 Bilal
(Cheema = slot 4 is 1100 and Bilal = slot 5 is 1200, so 1000 falls to Ali.)
The two discarded branches are the ones a hurried clerk skips; both die on rule 3, not on the rule that was just applied.
2. A cadet claims: 'Danish always flies before Cheema.' Decide whether this is necessarily true under the stated conditions, and distinguish this from merely being true of the published timetable.
Read the worked answer
It is necessarily true. In every surviving branch Danish is forced to slot 2:
- If Cheema-Bilal is (3,4), slot 4 is occupied and Danish must take the only other even slot, 2.
- If Cheema-Bilal is (4,5), slot 4 is again occupied and Danish takes 2.
- The (2,3) branch would have forced Danish to 4, but that branch already fails.
So Danish = 2 in the only consistent timetable, and Cheema = 4. Danish precedes Cheema by two slots in every model of the conditions, not just in the printed version.
The distinction that matters: a timetable is one witness, a logical consequence is a guarantee. Saying 'Danish flies before Cheema' because the published sheet shows it is an accident; saying it because no valid arrangement exists in which it is false is a deduction. In planning terms, only the second claim survives a change to the inputs - for example, if a slot were added.
3. The clerk suggests dropping one condition to make the timetable 'quicker to justify'. Determine whether any single condition can be removed without losing the unique solution, and state the consequence.
Read the worked answer
Test each in turn.
Remove rule 5 (Cheema not first): Cheema-Bilal could then be (1,2). Danish must be 4. Slots 3 and 5 go to Ali and Ejaz; Ali cannot take 5, so Ali = 3, Ejaz = 5 - but then Ejaz is later than Ali, breaking rule 4. So (1,2) still dies, and the solution is unchanged. Rule 5 does no work.
Remove rule 4 (Ejaz before Ali): with Cheema-Bilal = (4,5) and Danish = 2, slots 1 and 3 remain for Ali and Ejaz. Ali may be 3 with Ejaz at 1, or Ali may be... only 1 and 3 are free and Ali cannot take 1, so Ali is still forced to 3. Solution unchanged - but this must be shown, not assumed.
Remove rule 3 (Ali not first or last): branch (2,3) with Danish = 4 now puts Ali in slot 1 or 5. Two solutions appear. Uniqueness lost.
Remove rule 2 (Danish in an even slot): branch (2,3) puts Danish in 1 or 5 and Ali must avoid 1 and 5 - impossible; but branch (4,5) leaves Danish free at 1, 2 or 3, generating extra solutions. Uniqueness lost.
Remove rule 1: the pair constraint is the spine of the puzzle. Uniqueness lost.
Conclusion: exactly one condition - rule 5 - is redundant. Dropping any other destroys the unique answer, so the clerk's 'simplification' would either change nothing (rule 5) or change everything.