Question

There are five suburbs in the city of Melbourn. How many all-stars must be picked from each suburb to guarantee that at least five players come from the same suburb?

Short answer

Pick 21 players to guarantee at least five from one suburb.

Step-by-step solution

Understand the Problem

We need to find the minimum number of all-star players that must be selected from five different suburbs such that at least one suburb has a minimum of five players.

Worst-case Scenario Analysis

Consider the situation where the distribution of players is as even as possible across the five suburbs without any one suburb having 5 players. The maximum number of suburbs available is 5.

Distribute Maximum Players Evenly

In a worst-case scenario, each suburb can have 4 players without any having 5. This is calculated by choosing 4 players from each of the 5 suburbs, totaling 20 players (i.e., 4 players/suburb × 5 suburbs).

Calculate the Critical Number

The critical number is reached when we add one more player to this configuration. With 20 players distributed (4 in each suburb as calculated before), adding one more player forces the total in one suburb to become 5.

Conclusion

Therefore, to guarantee that at least one suburb has 5 players, you must pick 21 players (20 to evenly distribute and 1 additional to ensure one suburb reaches 5).

Key concepts

Pigeonhole Principle

The pigeonhole principle is a fundamental tool in combinatorics and mathematics. It states that if you have more items than containers to put them into, at least one container must hold more than one item. This principle helps in solving problems where you need to guarantee the distribution of items—or in this case, players—in a specific manner.
In the context of our problem, consider each suburb as a 'pigeonhole' and each all-star player as a 'pigeon'. By the pigeonhole principle, if we are trying to fit players (pigeons) into suburbs (pigeonholes), and we need at least one suburb to have a minimum of five players, we must ensure there are excess players beyond a simple one-per-suburb scenario. This principle guided us to determine the minimum necessary to satisfy the problem's condition.

Worst-case Scenario

The worst-case scenario analysis helps determine the maximum number of players we can distribute evenly without satisfying the condition of the exercise.
In the problem, the worst-case is when every suburb has an equal number of players but none reaches five. By distributing 4 players to each suburb (4 x 5 = 20 players), none of them contains 5 players, adhering to the worst-case distribution. This ensures that any new addition of a player will necessitate one suburb exceeding the threshold outlined in the problem.
Understanding the worst-case scenario is crucial because it helps in finding edge cases, where simply achieving the conditions of the problem can become complicated if not analyzed thoroughly.

Suburb Player Distribution

In our exercise, suburb player distribution is pivotal. Given that there are five suburbs in Melbourn, we must consider how to distribute players among them, to reach a specific outcome.
Initially, we distribute 4 players to each of the five suburbs, which results in 20 players. Here, the distribution is even, and none of the suburbs has reached the threshold of 5 players.
This type of distribution gives us insight into how close we are to achieving the desired condition, helping in strategizing further moves—especially in exercises involving similar distribution challenges where uniformity almost reaches the target, but one more step is needed to fulfill the requirement.

Minimum Number Selection

The concept of minimum number selection refers to identifying the least count of players needed to ensure that at least one suburb contains 5 players. By analyzing the distribution of players, we found that 21 is the critical number in our exercise.
After distributing 20 players equally, each suburb has 4. By selecting one more player—a total of 21 players—a suburb can finally reach the necessary count of 5 players. This guides us in solving similar problems where a minimum specific condition must be met among various groups.
Focusing on the minimum number selection ensures efficiency and precision in problem-solving, leaving no room for ambiguity in how the conditions of the exercise are satisfied.