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Maths olympiad question · P5–P6★★★★ Expert

Logic puzzles: olympiad question 11

An expert question on logic puzzles for P5–P6, with a step-by-step solution and the answer. The Hua Xia Cup's P4 qualifying round and finals have true-or-false statements and match-scoring questions. A favourite thinking workout, too.

The question

Yan (No.

7), Sze-ki (No.

6), Lok (No.

3), May (No.

5) run a 100 m race with no ties.

Before it, four adults each make two guesses:

Mr Chan:

"May third, Lok fourth."

Ms Lee:

"Lok first, Yan first."

Uncle Cheung the janitor:

"May fourth, Lok first."

Mrs Ho, a parent helper:

"May first, Sze-ki fourth."

Afterwards, each adult turns out to be right exactly once.

Write the numbers of the first to fourth placed, in order, as a four-digit number. What is it?

Show the hints (3)
  1. Hint 1: Start with Mr Chan: one guess right, one wrong. Take both cases, noting who is in which place and who isn't.
  2. Hint 2: In each case move to the next adult, again two ways; as soon as someone gets two places, a place gets two people, or a 'not' is broken, that case fails.
  3. Hint 3: In the end only one case fits everything: Lok is first in it.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Each adult is right exactly once: take the adults in turn, two cases each (which guess is right), note what follows, and strike out any case that clashes.
  2. 2.if Mr Chan's "May third" is right ("Lok fourth" wrong); if Ms Lee's "Lok first" is right ("Yan first" wrong); if Uncle Cheung the janitor's "May fourth" is right ("Lok first" wrong) — but May is already third: no.
  3. 3.if Mr Chan's "May third" is right ("Lok fourth" wrong); if Ms Lee's "Lok first" is right ("Yan first" wrong); if Uncle Cheung the janitor's "Lok first" is right ("May fourth" wrong); if Mrs Ho, a parent helper's "May first" is right ("Sze-ki fourth" wrong) — but May is already third: no.
  4. 4.if Mr Chan's "May third" is right ("Lok fourth" wrong); if Ms Lee's "Lok first" is right ("Yan first" wrong); if Uncle Cheung the janitor's "Lok first" is right ("May fourth" wrong); if Mrs Ho, a parent helper's "Sze-ki fourth" is right ("May first" wrong) — the rest must be Yan second; everything fits.
  5. 5.if Mr Chan's "May third" is right ("Lok fourth" wrong); if Ms Lee's "Yan first" is right ("Lok first" wrong); if Uncle Cheung the janitor's "May fourth" is right ("Lok first" wrong) — but May is already third: no.
  6. 6.if Mr Chan's "May third" is right ("Lok fourth" wrong); if Ms Lee's "Yan first" is right ("Lok first" wrong); if Uncle Cheung the janitor's "Lok first" is right ("May fourth" wrong) — but first is already Yan: no.
  7. 7.if Mr Chan's "Lok fourth" is right ("May third" wrong); if Ms Lee's "Lok first" is right ("Yan first" wrong) — but Lok is already fourth: no.
  8. 8.if Mr Chan's "Lok fourth" is right ("May third" wrong); if Ms Lee's "Yan first" is right ("Lok first" wrong); if Uncle Cheung the janitor's "May fourth" is right ("Lok first" wrong) — but fourth is already Lok: no.
  9. 9.if Mr Chan's "Lok fourth" is right ("May third" wrong); if Ms Lee's "Yan first" is right ("Lok first" wrong); if Uncle Cheung the janitor's "Lok first" is right ("May fourth" wrong) — but Lok is already fourth: no.
  10. 10.Only one case works: first Lok (No. 3), second Yan (No. 7), third May (No. 5), fourth Sze-ki (No. 6). The number is 3756.
  11. ⚠️Common mistake: deciding from the first two or three adults without checking the last one's guesses; or ignoring the wrong guess — it says someone is not in that place.

Answer: 3756

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