Sally was given a set of 5 cards numbered 1 to 5 and Peter was also given a set of 5 cards

Q: 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 guess the two numbers. Below is what each of them said:

Peter: I do not 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 told is larger than the number you were told.

Peter: Now I know the two numbers.
What are the two numbers?
A. 1 and 4
B. 1 and 5
C. 2 and 4
D. 2 and 5

Answer: A

Since each of them were each given numbers 1 to 5, if Peter was told any of the following numbers, he would be able to tell what the two numbers that were picked were:

  • 1 → 1 × 1 The two numbers are 1 and 1.
  • 2 → 1 × 2 The two numbers are 1 and 2.
  • 3 → 1 × 3 The two numbers are 1 and 3.
  • 5 → 1 × 5 The two numbers are 1 and 5.
  • 6 → 2 × 3 The two numbers are 2 and 3.
  • 7 → 1 × 7 The two numbers are 1 and 7.
  • 8 → 2 × 4 The two numbers are 2 and 4.
  • 9 → 3 × 3 The two numbers are 3 and 3.
  • 10 → 2 × 5 The two numbers are 2 and 5.
  • 12 → 3 × 4 The two numbers are 3 and 4.
  • 15 → 3 × 5 The two numbers are 3 and 5.
  • 16 → 4 × 4 The two numbers are 4 and 4.
  • 20 → 4 × 5 The two numbers are 4 and 5.
  • 25 → 5 × 5 The two numbers are 5 and 5.

(Products 11, 13, 14, 17, 18, 19, 21, 22, 23 and 24 cannot be formed.)

The only product that is ambiguous is 4 since 4 could be equal to 1 × 4 or 2 × 2. Therefore, when Peter said that he did not know the numbers, Sally would be able to know that the product Peter was told had to be 4. Since Sally said that the sum she was told is larger than the product Peter was told, the two numbers that were picked had to be 1 and 4 (sum = 5) and not 2 and 2 (sum = 4).