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 have 5 integers.
To explain how this is possible, we need to understand what it means for a number to be "third in order" when arranging a set of integers from least to greatest and from greatest to least.
When arranging a set of numbers from least to greatest, the number that is "third in order" means that there are two numbers that are smaller or equal in value that come before it.
Similarly, when arranging a set of numbers from greatest to least, the number that is "third in order" means that there are two numbers that are greater or equal in value that come before it.
So, in Ted's case, there exists a number in the set that satisfies the condition of being the third in order, both when arranging the integers from least to greatest and from greatest to least.
Let's consider an example to understand this better. Suppose Ted has the following set of integers: {7, 3, 5, 1, 4, 6, 2}.
When arranging these numbers from least to greatest, in order: 1, 2, 3, 4, 5, 6, 7, we can see that the number "3" is third in the order.
When arranging these numbers from greatest to least, in order: 7, 6, 5, 4, 3, 2, 1, we can see that again, the number "3" is third in the order.
Therefore, the existence of a number that is third in order in both arrangements is possible because the position of numbers changes when arranging them from least to greatest or greatest to least.