IMC 2013 Q23
23. Four brothers are discussing the order in which they were born. Two are lying and two are telling the truth. Which two are telling the truth?
- A Bernard and Inigo
- B Horatio and Bernard
- C Alfred and Horatio
- D Alfred and Bernard
- E Inigo and Horatio
Intermediate Mathematical Challenge 2013, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A Bernard and Inigo
If Alfred is telling the truth, the other three are lying (as their statements would then be false) and we know this is not the case. Hence Alfred is lying. Similarly, if Horatio is telling the truth, the other three are lying which again cannot be the case. So Horatio is lying. Hence the two who are telling the truth are Bernard and Inigo. (A case where this situation would be realised would be if the brothers in descending order of age were Alfred , Bernard, Horatio and Inigo.)
Solution © UK Mathematics Trust, from the IMC 2013 solutions. Longer versions with investigations: Solutions and investigations (PDF).