17.6. Question 9

Here you are given the following. You are choosing 5 integers from the set 1 to 8 arbitrarily. Our goal is to show irrespective of the way you choose those 5 points there always exists at least one pair of integers among those chosen 5 integers whose sum is 9.

In this section, we have a task where we need to choose 5 integers from a set of integers ranging from 1 to 8. The aim is to prove that regardless of how we pick these 5 integers, there will always be at least one pair of numbers in our selection that adds up to 9. For example, if we choose the numbers 1, 2, and 5, and then include 3, we do not yet have a pair that sums to 9. But once we add 4, we get 5 + 4 = 9, which satisfies our condition.

Think of a fruit basket with 8 types of fruits, numbered from 1 to 8. If you randomly pick 5 fruits, the required condition is akin to ensuring that among the choices, you can always find two fruits that together weigh exactly 9 grams. Just as you can be confident that within those 5 diverse fruits, there's a pair that meets the weight requirement, in our selection of integers, there's always a pair that will sum to 9.

Now it follows simply from the pigeonhole principle that there always exists a pair of values out of the 5 numbers.

The pigeonhole principle states that if you have more items than containers to put them in, at least one container must hold more than one item. Here, our 5 chosen integers are like 'pigeons,' and the possible pairs of numbers that add up to 9 are the 'holes.' For our set of integers {1, 2, 3, 4, 5, 6, 7, 8}, the unique pairs that sum to 9 are (1,8), (2,7), (3,6), and (4,5). Since there are only 4 pairs but we are selecting 5 integers, the principle ensures that at least two integers must pair up to make 9.

Imagine you have 5 different colored marbles and only 4 boxes to place them in, where each box can only hold one specific color. If you try to store all 5 marbles, one of the boxes will have to hold at least two marbles. Similarly, in our case, picking 5 integers ensures at least one pair will sum to 9 since we only have 4 unique sum-pairs available.

So, without loss of generality, suppose both got mapped to (1, 8); we do not know what is the identity of x1 or x2.

Here the proof concludes by assuming a specific case where two of the integers selected map to the pair (1, 8) under our defined mapping. Regardless of which integers these are, their identities do not change the outcome; they will still sum to 9. This 'without loss of generality' approach means that the specific integers chosen do not matter, as the principle confirms a similar outcome across all combinations of selections.

Imagine you are running a race but don't know which runners will finish first and second. Regardless of the uniforms they wear (analogous to the integer identities), the first two finishers will always bag the same gold and silver medals. This is akin to our mathematical proof that leads us to conclude that one of the pairs selected will always reach the 'finish line' of summing to 9.

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