IMC 2014 Q11
11. Not all characters in the Woodentops series tell the truth. When Mr Plod asked them, “How many people are there in the Woodentops family?”, four of them replied as follows:
Mrs Scrubitt: “A number which is the product of two integers greater than one.”
How many of these four were telling the truth?
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2014, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 2
Precisely one of Jenny and Willie is telling the truth since the number of people is either even or odd. Similarly, precisely one of Sam and Mrs Scrubitt is telling the truth since the number of people is either a prime number or a number which is the product of two integers greater than one. So although it is not possible to deduce who is telling the truth, it is possible to deduce that exactly two of them are doing so.
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).