Four Riders, Four Constraints
Four cyclists - Hana, Ivo, Jun and Kira - finished a race in four distinct positions, 1st to 4th. The following are known:
(i) Ivo finished neither first nor last. (ii) Jun finished immediately before Ivo. (iii) Hana did not finish last. (iv) The race was not won by Jun.
No two riders shared a position. Every question below must be answered from these four statements alone; no outside knowledge of the riders is relevant.
1. Determine the finishing order from 1st to 4th, showing how the constraints interact.
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Start with the most restrictive constraint, (i): Ivo is 2nd or 3rd.
Case Ivo = 2nd: by (ii), Jun is 1st. But (iv) says Jun did not win, so this case is impossible.
Case Ivo = 3rd: by (ii), Jun is 2nd. Positions 1 and 4 remain for Hana and Kira. By (iii), Hana is not last, so Hana is 1st and Kira is 4th.
All four constraints hold: Ivo 3rd (not first or last ✓), Jun immediately before Ivo ✓, Hana not last ✓, Jun did not win ✓.
Order: Hana 1st, Jun 2nd, Ivo 3rd, Kira 4th.
Method point: test the constraint that creates a small number of branches first, then eliminate branches with the 'negative' constraints - that is faster and safer than guessing an order and checking it.
2. If constraint (iv) were removed, how many different valid finishing orders would exist, and what does that tell you?
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Re-run both branches with only (i), (ii) and (iii):
- Ivo 2nd → Jun 1st. Remaining: Hana and Kira in 3rd and 4th. Hana cannot be last, so Hana 3rd, Kira 4th. 1 order.
- Ivo 3rd → Jun 2nd. Remaining: Hana and Kira in 1st and 4th. Hana cannot be last, so Hana 1st, Kira 4th. 1 order.
Total: 2 valid orders.
What this tells you: Kira is 4th in both, so her position is forced by (i)–(iii) alone and (iv) adds nothing about her. Constraint (iv) does exactly one job - it separates the two remaining orders by ruling out Jun winning. If an exam question asks 'which constraint is essential to reach a unique answer', the answer is (iv), not any of the others.
3. Suppose you are told separately that Ivo finished second. Does a consistent arrangement exist under all four constraints? Explain your reasoning.
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No consistent arrangement exists.
Ivo 2nd forces Jun 1st by (ii), because 'immediately before' with Ivo in position 2 places Jun in position 1. But (iv) states Jun did not win.
So the new information directly contradicts (iv). This is the key skill being tested: not 'find an order that fits the new fact' but 'check the new fact against every constraint before generating an order'. A candidate who stops at 'Ivo 2nd, Jun 1st, then Hana and Kira in 3rd and 4th' has produced an arrangement that violates a stated rule and must be rejected.
Sanity check with (iii): Hana cannot be last, so the only would-be arrangement is Hana 3rd, Kira 4th - that satisfies (i), (ii) and (iii) but still breaks (iv). One broken constraint is enough to make the scenario impossible.