A hard question on logic puzzles for P4–P5, with a step-by-step solution, diagrams 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
In maths, the teacher thinks of a two-digit number and lets the class guess its features.
4 pupils each say one thing.
Keung:
"Divided by 5, it leaves remainder 1."
Sze-ki:
"Its two digits add up to 7."
Tsz-hin:
"Its two digits multiply to 6."
Ming:
"It's smaller than 55."
The teacher says, "Exactly one of you is wrong."
What two-digit number is the teacher thinking of?
Show the hints (3)
💡Hint 1: Only one is wrong. If you knew who, everything else would be true.
💡Hint 2: Try each person in turn as the one who's wrong. Each time, list the two-digit numbers fitting the others; there won't be many.
💡Hint 3: Careful: if a number you find also fits the words of the one who's "wrong", then that person isn't wrong, and the case fails.
Worked solution
Have a go first, then read the solution step by step.
1.Exactly one is wrong; try each person as that one.
2.If Keung is the one who's wrong, the other 3 are right. The numbers fitting them: 16. But 16 also fits Keung's words, so Keung wouldn't be wrong — no.
3.If Sze-ki is the one who's wrong, the other 3 are right. The numbers fitting them: 16. But 16 also fits Sze-ki's words, so Sze-ki wouldn't be wrong — no.
4.If Tsz-hin is the one who's wrong, the other 3 are right. The numbers fitting them: 16. But 16 also fits Tsz-hin's words, so Tsz-hin wouldn't be wrong — no.
5.If Ming is the one who's wrong, the other 3 are right. The numbers fitting them: 16, 61. But 16 also fits Ming's words, so Ming wouldn't be wrong. 61 doesn't fit Ming's words, so Ming really is wrong: this works.
6.Only one case works: Ming is wrong, and the number is 61.
⚠️Common mistake: stopping once a number fits the others, without checking that it really makes the "wrong" person wrong.
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download