Congratulations to Manjit from the main office for submitting a correct solution!
Solution:
Let's first list all possible sets of three cards with different numbers in increasing order having a sum of 13:
(1, 2, 10)(1, 3, 9)(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)(2, 5, 6)(3, 4, 6)
When Alyssa looked at the card on the left but could not tell what the other two cards were, we knew she was not looking at “3” since 3-4-6 is a unique solution and she would know the other two cards in that case. So we can rule out the last case.
(1, 2, 10)(1, 3, 9)(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)(2, 5, 6)
Brandon then looked at the rightmost card and said he could not tell what the other two cards were. By the same reasoning, we can rule out 1-2-10, 1-3-9, and 2-5-6 because each has a unique rightmost card.
(1, 4, 8)(1, 5, 7)(2, 3, 8)(2, 4, 7)
When Christina looked at the middle card and said she could not tell what the other two cards were, by the same reasoning, we can rule out 1-5-7 and 2-3-8 since 5 and 3 are unique numbers in the middle. That leaves us 4 as the only possible middle number. So the answer is 4.
(1, 4, 8)(2, 4, 7)
