Five-Instrument Calibration Week: Testing Whether a Schedule Still Holds
A calibration laboratory must certify five instruments - A, B, C, D and E - before a test campaign. Each instrument needs one full working day, only one instrument can be handled per day, and the week available runs Monday to Friday. After checking technician availability the in-charge fixed five conditions: (i) C cannot be taken on Monday or on Friday; (ii) A must be taken on the day immediately before E; (iii) B must be completed before C; (iv) D cannot be taken on Wednesday; (v) E must be taken on Thursday. On Tuesday morning the store reports that instrument C has gone out for repair and can only be taken on Friday. The campaign manager wants a straight answer on whether the week still works and whether the days of any other instruments must shift.
1. Using conditions (i) to (v), work out the original Monday-to-Friday order and justify each placement.
Read the worked answer
Start from the most restrictive condition, which is (v).
- E = Thursday (condition v).
- A = Wednesday, because (ii) forces A to be the day immediately before E and Thursday's predecessor is Wednesday.
- Free slots left: Monday, Tuesday, Friday, for B, C, D.
- Condition (i) bars C from Monday and Friday, so C = Tuesday.
- Condition (iii) requires B before C, and Tuesday is C's day, so B = Monday.
- The only remaining slot is Friday, so D = Friday. Check condition (iv): D is not on Wednesday - satisfied.
Final order: Monday B, Tuesday C, Wednesday A, Thursday E, Friday D.
This order is unique, not merely one of several: E's fixed day pins A, and the C restriction then pins everything else. Any other reading would put B after C or C on a barred day.
2. With C forced to Friday, is the week still workable? Identify precisely which condition fails and whether any other instrument must move.
Read the worked answer
Re-run the deductions with C on Friday.
- Condition (v): E = Thursday still holds.
- Condition (ii): A = Wednesday still holds.
- Free slots: Monday and Tuesday, to be filled by B and D.
- Condition (iii), B before C: satisfied on either Monday or Tuesday, since C is Friday.
- Condition (iv), D not on Wednesday: satisfied on either Monday or Tuesday.
So the week is workable, but only by relaxing condition (i), the ban on C being taken on Friday. That is the single condition that fails; E and A keep their days and no other instrument is forced to move.
Important caveat for the manager: two orders now satisfy every surviving condition, namely B-Mon/D-Tue and D-Mon/B-Tue. If the order matters - for example, if D's result must feed A's calibration certificate - a tie-breaker must be added deliberately, otherwise two technicians may produce two different plans.
3. Suppose the lab chief instead insists that E be taken on Wednesday. Does a fully compliant order exist? Show it or prove it cannot exist.
Read the worked answer
Test it rather than assume it.
- E = Wednesday, so by (ii) A = Tuesday.
- Remaining slots: Monday, Thursday, Friday for B, C, D.
- Condition (i) allows C only on Thursday (Monday and Friday are barred; Tuesday and Wednesday are taken).
- Condition (iii) then forces B before Thursday, and Monday is the only free slot before it, so B = Monday.
- D = Friday, which satisfies (iv) because it is not Wednesday.
Result: Monday B, Tuesday A, Wednesday E, Thursday C, Friday D - fully compliant with all five conditions.
So yes, an order exists. The useful lesson is that condition (v) as written (E on Thursday) is stricter than the problem actually needs; shifting E one day earlier absorbs the Friday repair slot without breaking anything. A candidate who simply says 'impossible' has not tested the alternatives, and a planner who renegotiates E's day gets a compliant week while the C repair is pending.