Debate Tournament Seeding
Five teams-Green, Red, Blue, Yellow, White-finish a debate tournament in distinct places 1 through 5. The tabulation sheet is lost. The coach remembers four clues: Green finished higher than Red, and Red higher than Blue; Yellow finished immediately above White; Blue did not finish last; White finished lower than Blue. A student claims the order must be Green, Red, Blue, Yellow, White. Another student says the clues allow at least two different orders. The coach has one additional memory: “The team that finished third was not White.” The group must decide the full order, test whether the student’s order is forced, and identify any clue that adds nothing once the others are known. The coach wants a written explanation, not just a ranking, because the league requires a defensible tiebreak record.
1. Determine the full finishing order. Show the forcing step that eliminates alternatives.
Read the worked answer
Order: Green, Red, Blue, Yellow, White. Yellow must sit immediately above White, so that pair uses two consecutive places. White is lower than Blue, so Blue must be above White; if Blue were fourth, White would have to be fifth, but then Yellow would also need to be fourth-impossible. If Blue were fifth, White would have to be lower than fifth-impossible. Therefore Blue is at most third. Since Green > Red > Blue, the only fit is Green first, Red second, Blue third. The remaining places are fourth and fifth, and the consecutive pair Yellow–White fills them in that order. The alternative “at least two orders” claim fails because every other placement puts White below Blue without room for the Yellow–White pair.
2. Is the student’s proposed order forced? Explain what “forced” means here.
Read the worked answer
Yes. “Forced” means no other assignment satisfies all clues at once. The key constraint is not the student’s ranking itself but the interaction of two clues: Yellow immediately above White and White lower than Blue. Those two clues, together with Green > Red > Blue, pin Blue to third. Once Blue is third, Green and Red must occupy first and second in that order, and Yellow–White must occupy fourth and fifth. A valid explanation should test a rival order-for example, Green, Blue, Red, Yellow, White-and show that it violates Red > Blue.
3. Which clue is redundant, and why?
Read the worked answer
“Blue did not finish last” is redundant. If White finished lower than Blue, then Blue cannot be last: there must be at least one team below Blue, namely White. So the “not last” clue adds no new restriction. The coach’s extra memory that third place was not White is also redundant, but for a different reason-the main clues already force Blue into third. A good answer distinguishes redundancy from mere repetition: redundancy means removing the clue leaves the same solution set.