Meridian Labs: sequencing five induction sessions
Meridian Labs runs a one-day induction for 40 new engineering hires, and the day has exactly five 75-minute sessions in five slots: 9:00, 10:30, 12:00, 2:00 and 3:30. The sessions are Product (P), Pricing (R), Legal (L), Sales (S) and Ops (O), one per slot. The scheduling rules agreed with the facilitators are: Product must be immediately followed by Pricing in the next slot; Legal cannot be in slot 1 or slot 5, because the external counsel joins late and leaves early; Sales must come somewhere before Ops; and Pricing cannot be in slot 5. The CEO will open the day with a ten-minute welcome note, which is not counted as one of the five sessions. The programme manager has drafted a schedule with Ops at 12:00 and wants to know whether it can survive the rules. A facilitator has also asked whether slot 5 can be left free for an open house; the manager suspects not, but cannot say why.
1. Which session, if any, is forced into slot 5? Justify the answer from the rules rather than by listing schedules.
Read the worked answer
Ops is forced into slot 5.
Work from the Product–Pricing block. Because Product must be immediately followed by Pricing and Pricing cannot be in slot 5, the block can occupy only (1,2), (2,3) or (3,4) - never (4,5). Three slots therefore remain for Legal, Sales and Ops, and whichever position the block takes, slot 5 is one of those three remaining slots: block at (1,2) leaves 3, 4, 5; block at (2,3) leaves 1, 4, 5; block at (3,4) leaves 1, 2, 5.
Now eliminate the other candidates for slot 5. Legal is barred from slot 5 by rule. Sales must come before Ops, so Sales cannot be last. That leaves Ops as the only session that can occupy slot 5.
This also disposes of the manager's draft: Ops at 12:00 (slot 3) is impossible, because it would leave slot 5 to Product, Pricing, Legal or Sales, and every one of those is excluded by a rule.
2. If Legal is fixed at slot 4, how many valid schedules exist and what are they?
Read the worked answer
With Legal at slot 4, the Product–Pricing block can sit only at (1,2) or (2,3): (3,4) is taken by Legal, and (4,5) is banned because Pricing cannot be last.
- Block at (1,2): slots 3 and 5 remain for Sales and Ops; Sales must precede Ops, so Sales = 3, Ops = 5. Schedule: P, R, S, L, O.
- Block at (2,3): slots 1 and 5 remain for Sales and Ops; Sales = 1, Ops = 5. Schedule: S, P, R, L, O.
Exactly two valid schedules, and both place Ops in slot 5 - an independent confirmation of the forced conclusion rather than a coincidence. Note also that neither schedule can be rearranged to put Ops earlier: in each case the remaining slots are consumed by the block and by Sales, which must precede Ops.
3. The facilitator wants slot 5 kept free for an open house. What is the correct response, and what alternatives are genuinely open?
Read the worked answer
Slot 5 cannot be freed while all five sessions must be scheduled. The proof above shows Ops has no other legal home: Legal is barred from slot 5 and from slot 1, and Sales cannot be last, so removing Ops from slot 5 leaves a slot that no remaining session may occupy. Keeping the slot free would require relaxing one of the stated rules, not merely reshuffling.
The honest options, in order of cost:
- Move the open house outside the five slots - before 9:00 or after the 3:30 session ends at 4:45. Costs nothing and breaks no rule.
- Relax the Pricing-not-last rule to allow the block at (4,5); this frees a different slot but changes what the facilitators agreed to, and needs the Pricing lead's consent.
- Relax the Legal constraint, which is externally imposed by counsel's availability and is the least negotiable of the four.
What should not be done is to promise the facilitator the open house at 3:30 on the assumption that 'it can probably be worked out'. The constraint set has a unique structural answer, and the manager can state it with certainty.