Mathematical genius Adam Spencer is the author of a new book The Number Games. Spencer is sharing a daily riddle designed to get your grey matter firing in the lead up to the festive season.
This puzzle's a little hard, but have a go.
Sally was given a set of 5 cards numbered 1 to 5, and Peter was also given a set of 5 cards numbered 1 to 5. They were then blindfolded and told to pick a card from their respective sets.
The sum of the numbers from the two cards was told only to Sally and the product of the numbers was told only to Peter. They were then told to name the numbers on the two cards they had chosen. Below is what each of them said.
Peter: I don't know the two numbers.
Sally: Now I know the two numbers.
Peter: I still don't know the two numbers.
Sally: Let me help you. The number I was just told is larger than the number you were just told.
Peter: Now I know the two numbers!
So, what were the two numbers?
Peter has been told the product of the two numbers. But he doesn't know which numbers give him that product. What does that tell us about the product and the possible numbers involved?
The crucial thing to realise here is that even when people are answering "I don't know" they are actually giving us important information. Look at it this way; if Peter was told the product was 5 he would know the cards have to be 1 and 5.
If he was told the product was 12, it could only be 3 and 4. There is only one product that could leave him unsure and that is 4, because it could be 1 and 4 or 2 and 2.
So by saying "I don't know" Peter is actually telling Sally that the product he was told was 4. So now Sally knows it has to be 1 and 4 or 2 and 2. And she has already been told the sum. So Sally knows the cards are 1 and 4. If she tells Peter the sum is larger than the product, Peter also knows the cards are 1 and 4.
Adam Spencer's book The Number Games is available now from all good bookstores or visit adamspencer.com.au. Note: Shipping to NZ not available.