Yesterday I bought two tickets to a movie, which logically had consecutive numbers. I examined the numbers, which had four digits, and mentioned that the sum of the eight digits was 25.
A friend asked if any digit appeared more than twice out of the eight, and I answered him.
Other friend asked if the sum of the digits of either ticket was equal to 13, which I also answered.
And then, to my surprise, my daughter told me what the two numbers were!
What were they?
ANSWER TO THE TWO TICKET PROBLEM:
We must make the assumption that the two tickets are sequential in number such that the first is ABCD and the second ABC(D + 1) with some minor possible modifications as discussed later. Given that the tickets were purchased at the same time, the sequential sequence seems reasonable.
Scenario 1: Let the first ticket be ABCD where D is less than 9. Then the second ticket is ABC(D + 1). Then the total of the two tickets is 2A + 2B + 2C + 2D + 1 = 25. Then one ticket sums to 12 and the higher number ticket sums to 13. However, there are so many numbers that meet this criterion that there is no way we could find a unique answer. Then we know that the answer to the question "Does either ticket sum to 13?" is a resounding "No".
Scenario 2: Let the first ticket be ABC9 and the next be AB(C + 1)0. Then 2A + 2B + 2C + 9 + 1 = 25. Then 2 x (A + B + C) = 15. Then A + B + C = 7.5. Since each letter represents an integer, we can dispense with this scenario.
Scenario 3: Revise the second scenario such that the first ticket is AB99 and the second ticket A(B + 1)00. Then 2A + 2B + 19 = 25. So 2 x (A + B) = 6 so A + B = 3. Then we have four scenarios:
1 First ticket 0399 second ticket 0400
2 First ticket 1299 second ticket 1300
3 First ticket 2199 second ticket 2200
4 First ticket 3099 second ticket 3100
We can see that the first, third and fourth scenarios each have a specific digit that occurs more than twice. Therefore, the correct ticket combination is the second: 1299 and 1300. We can see that the answer to each of the two questions ("Does either ticket sum to 13?", and "Does either ticket have a number that occurs more than twice?" must be NO!
I like this problem, as it provides such a large number of possibilites initially and yet narrows down eventually to a unique answer.
Gordon S.
Answer to the Two Ticket Problem
