Question

Carol holds a two-digit number X and wants Alice and Bob to recover the number. She gives them the following information.

• Both Alice and Bob know that the number held by Carol is one of the following.

• 21  26
• 31  32  35
• 42  44
• 52  54  57
• 62  67

• Alice knows the number of ones of X but not the number of tens.
• Bob knows the number of tens of X but not the number of ones.

Alice and Bob exchange the following messages.

1. (a)  Bob to Alice: ’You might know X’.
2. (b)  Alice to Bob: ’I did not, but, after what you have said, I do’.

Based on the above information, please answer the following questions. For each question, please accompany your answer with a short explanation of why your believe your answer is correct. Marks will not be awarded for answers without any explanation.

1. (a)  Based on the statement of Bob it is possible to narrow down candidates for the number of tens from the initial five possibilities down to just two. What are these two possibilities? (7 marks)
2. (b)  Based on the ’I did not’ statement of Alice, the number of candidates for X can be narrowed down to just three. What are these candidates (6 marks)
3. (c)  The remaining part of Alice’s statement allows to identify X precisely. What is X? (7 marks)