Carl thought of 5 numbers and told Danny all the sums that could be made by adding the numbers in pairs. The list of sums is given below. What are the five numbers?
6,7,8,8,9,9,10,10,11,12
I thought of 34426 but it doesn't give the 11
To find the five numbers that Carl thought of, we can use the given information about the sums of the numbers in pairs. Let's break down the steps to solve this:
1. Start by organizing the list of sums in ascending order: 6, 7, 8, 8, 9, 9, 10, 10, 11, 12.
2. Assume the five numbers are a, b, c, d, and e. We need to find these numbers.
3. From the list of sums, we see that the sum 6 can only be obtained by adding the smallest two numbers (a + b).
4. Similarly, the sum 12 can only be obtained by adding the largest two numbers (d + e).
5. Based on the list of sums, we see that the repeated sums (8, 9, and 10) indicate that some numbers are repeated in the set.
6. Since there are two occurrences of the sum 8, it means that two different pairs of numbers can be added to get 8. So, we have a few possibilities: (a + c), (b + d), or (b + e).
7. Similarly, since there are two occurrences each of the sums 9 and 10, we have the following possibilities:
- (a + d), (c + e) for 9.
- (a + e), (c + d) for 10.
8. Finally, based on the given list of sums, we know that (a + e) must equal 11.
Now we can use this information to form a system of equations and solve for the variables.
Let's consider each possibility:
Possibility 1: (a + c) = 8 and (b + d) = 8.
From this, we can conclude that a = 3, c = 5, b = 3, and d = 5. However, this possibility doesn't satisfy the equation (a + e) = 11.
Possibility 2: (a + d) = 9 and (c + e) = 9.
From this, we can conclude that a = 3, d = 6, c = 3, and e = 6. However, this possibility doesn't satisfy the equation (a + e) = 11.
Possibility 3: (a + e) = 11.
From this, we can conclude that a = 3 and e = 8. Since we already have two occurrences of the sum 8, we can conclude that c = 5. Now we need to find b and d such that (b + d) = 8. The remaining numbers are 4 and 1, and we can assign them to b and d in any order. So, the final set of numbers is {3, 4, 5, 1, 8}, which gives us all the given sums.
Therefore, the five numbers that Carl thought of are 3, 4, 5, 1, and 8.