Five Project Presentations: A Scheduling Puzzle
Five Grade 12 students - Abel, Bontu, Chala, Desta and Eyerusalem - must each present a project on one day of a review week, Monday to Friday, one presentation per day. The rules the coordinator wrote on the board are: Bontu presents after Chala but before Desta; Eyerusalem presents on Thursday; Abel does not present on Monday; Desta does not present on Friday. A classmate says: 'Since Eyerusalem is Thursday, Abel must be Friday - it's obvious.' Later the coordinator adds one more condition: Abel must present before Chala. Nothing else is fixed, and the coordinator insists that all five conditions must hold exactly as written.
1. Determine the full order of presentations. Is the classmate's conclusion correct, and is the reasoning valid?
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The order is forced: Monday Chala, Tuesday Bontu, Wednesday Desta, Thursday Eyerusalem, Friday Abel.
Reasoning. Eyerusalem has Thursday, leaving Monday, Tuesday, Wednesday and Friday for the other four. Chala, Bontu and Desta must appear in that order, so Desta cannot be on Friday. Consider where the triple Chala–Bontu–Desta can sit among Monday, Tuesday, Wednesday and Friday:
- Monday, Tuesday, Friday: Desta on Friday - impossible.
- Monday, Wednesday, Friday: Desta on Friday - impossible.
- Tuesday, Wednesday, Friday: Desta on Friday - impossible.
- Monday, Tuesday, Wednesday: Desta on Wednesday - possible.
The only remaining option is the first three days, with Abel on Friday. This also satisfies 'Abel does not present on Monday'.
So the classmate's conclusion is right but the reasoning is not. 'Eyerusalem is Thursday' by itself tells us nothing about Abel; the conclusion only follows once you also use the Chala–Bontu–Desta chain and the rule that Desta is not on Friday. Stating the right answer from an incomplete argument earns no method marks in a reasoning item.
2. Is the Thursday clue necessary? What happens to the number of possible orders if 'Eyerusalem presents on Thursday' is removed?
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Yes, it is necessary: without it the schedule is not unique.
Keep Chala before Bontu before Desta, Abel not Monday, Desta not Friday, and let Eyerusalem float. The triple C–B–D can occupy several sets of days:
- Monday, Tuesday, Wednesday, with Desta on Wednesday: Abel and Eyerusalem swap the remaining two days, so there are two orders.
- Monday, Tuesday, Thursday: Desta on Thursday; Abel can be Wednesday or Friday, while Eyerusalem takes the other - again valid orders.
- Monday, Wednesday, Thursday: also valid, with Abel on Tuesday or Friday.
- Tuesday, Wednesday, Thursday: Desta on Thursday, so Abel cannot be Monday; he must take Friday and Eyerusalem takes Monday - one order.
In total seven distinct orders satisfy the four remaining rules. For example, both 'Chala Monday, Bontu Tuesday, Desta Wednesday, Eyerusalem Friday, Abel Thursday' and 'Chala Monday, Bontu Tuesday, Desta Wednesday, Abel Friday, Eyerusalem Thursday' fit the other rules.
So a single extra fact - Eyerusalem on Thursday - is what removes the ambiguity. When asked whether a puzzle is 'soluble', the right check is not whether an answer can be produced, but whether exactly one arrangement survives all the constraints.
3. The coordinator now adds: 'Abel must present before Chala.' Show what happens, and advise the coordinator.
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The five rules together are contradictory, so no schedule exists.
Proof. From 'Abel before Chala before Bontu before Desta', these four students occupy four different days in strict order. Desta is not on Friday, so Desta is at latest on Thursday, which means the earliest possible block for the four is Monday to Thursday with Abel on Monday, Chala on Tuesday, Bontu on Wednesday and Desta on Thursday. Any later start would push Desta onto Friday. Abel is therefore forced onto Monday - which directly contradicts 'Abel does not present on Monday'. In fact the contradiction arises even before Eyerusalem is considered.
Advice. Something must be renegotiated, and the coordinator should be told which rules conflict and what the cheapest repair is. One consistent set is: drop 'Abel does not present on Monday', keep 'Abel before Chala', keep 'Bontu before Desta' and 'Desta not Friday', and move Eyerusalem to Friday. That gives Abel Monday, Chala Tuesday, Bontu Wednesday, Desta Thursday, Eyerusalem Friday. The coordinator should not simply announce a schedule and hope nobody checks: an impossible set of rules produces complaints on the day, and the correct response is to identify the pair of rules that cannot both stand.