18.2. Disjoint Group Sums

In this question we want to show that irrespective of what exactly are their ages, as long as they are in the range 18 to 58, it is always possible to choose 2 disjoint groups of people out of these 9 people whose sum of ages is the same.

The objective of this section is to prove that when you have 9 individuals aged between 18 and 58, you can always find 2 different groups among them that have the same total age. It's important to note that the two groups must be disjoint, meaning they cannot have any individuals in common.

Imagine 9 friends at a party, each having different ages between 18 and 58. If you want to split them into two teams for a game, the task is to ensure that the total age of each team is the same, even if the teams consist of different individuals.

We will do this by pigeonhole principle. So the first thing is since we want to argue about a non-empty set of people because when I want to consider the age of the people there have to be people in the group. So I have to focus on non-empty subsets. So if I have 9 people then the number of non-empty groups that I can form out of those 9 people need not be disjoint is 511.

To understand how many combinations of groups can be made from these 9 people, we use the concept of subsets. Each of the 9 people can either be included or not included in a subset, leading to a total of 2^9 or 512 possible subsets. However, since the empty set (where no one is selected) does not count, we subtract this, leading to 511 non-empty subsets.

Consider each person as a light switch that can be either on or off. Each combination of switches being on represents a different team composition. Since there are 9 switches, the total combinations of groups (or states) is 2^9. If everyone is 'on', you have a full group, and if all are 'off', you have the empty group, which you ignore for the count.

If I consider the minimum sum of ages possible in a group it could be 18. This is possible only when I have a group of just one person and that person has age 18. Whereas the maximum possible sum can occur when in my group I have all the 9 people and each of them has age 58. That means the range of possible sums here is 505.

When forming groups, the smallest sum occurs when the youngest person, aged 18, is included alone, summing to 18. The largest sum happens when all 9 are included at the maximum age of 58, resulting in 9 * 58 = 522, and we determine that the maximum range must account for every possible sum from 18 up to 522, giving a total of 505 possible sums.

Think of creating different age groups for a team, where the youngest player might bring only a score of 18 if they play alone, while having your oldest players together would yield the highest score possible. The range of potential total scores thus reflects all combinations of player ages.

So by pigeonhole principle, I can say that there always exists a pair of groups such that the sum of ages of the people in the groups is the same. But my question wants me to show that the groups should be disjoint.

Having more subsets (511) than possible sums (505) means, according to the pigeonhole principle, at least two subsets must share the same sum of ages. However, we need to ensure these groups are disjoint. If two selected groups share members, removing the common individuals will not alter their sums, leading to a valid solution of disjoint groups.

Imagine you have 511 different flavor combinations for ice cream (your subsets), and only 505 flavor profiles available (the sums). With so many combinations, it’s inevitable that at least two will taste the same. But if you have some friends to share, just scoop away common flavors to ensure everyone has their unique mix!

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