Ted put several different integers in order from least to greatest. He then put the same integers in order from greatest to least. Both times the same number was third in order. Explain how this is possible.
He must be using 5 different integers.
To understand how this is possible, let's break down the situation. Ted put several different integers in order from least to greatest, and then in reverse order from greatest to least. In both orderings, the same number was in the third position.
Let's consider a simple example to illustrate this scenario. Suppose Ted had the following set of integers: {5, 2, 7, 9, 1}.
Ted first arranges them in ascending order:
1, 2, 5, 7, 9
Here, the number 5 is in the third position.
Next, Ted arranges the same integers in descending order:
9, 7, 5, 2, 1
Again, we can see that the number 5 appears in the third position.
So, how is this possible?
The key to this scenario lies in the fact that there are multiple integers that occupy the same value. In this case, Ted had multiple occurrences of the number 5.
When Ted arranges the numbers in ascending order, the first occurrence of 5 is in the third position. However, when he rearranges the same set of numbers in descending order, the second occurrence of 5 takes the same third position.
In general, this situation is only possible when there are duplicate values in the set of integers. By having multiple occurrences of the same number, Ted was able to create two unique orderings where the third position was occupied by the same number.