Problem 62 - Discrete Mathematics with Applications

Question

A company wishes to schedule 1 -hour meetings beginning at 7 \(\mathrm{AM}\) . between every two of its six regional managers-A, B, C, D, and \(\mathrm{F}-\) so each can spend an hour with each of the other five for better acquaintance. Find the various possible schedule pairings. (This is a round-robin tournament in disguise.) (S. W. Golomb, 1993)

Step-by-Step Solution

Answer

The schedule for the 1-hour meetings between the six regional managers can be arranged in 5 time slots as follows: _Time Slot 1: 7 AM_ - Meeting 1: A with B - Meeting 2: C with D - Meeting 3: E with F _Time Slot 2: 8 AM_ - Meeting 1: A with C - Meeting 2: B with D - Meeting 3: E with F _Time Slot 3: 9 AM_ - Meeting 1: A with D - Meeting 2: B with E - Meeting 3: C with F _Time Slot 4: 10 AM_ - Meeting 1: A with E - Meeting 2: B with F - Meeting 3: C with D _Time Slot 5: 11 AM_ - Meeting 1: A with F - Meeting 2: B with C - Meeting 3: D with E This schedule ensures that each manager has a one-on-one meeting with the other five managers without any overlapping meetings.

Step 1: List all possible pairings

First, create a list with all possible pairings. In this case, the list of pairings can be represented in the following manner: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF.

Step 2: Distribute the pairings into time slots

Next, distribute the pairings over time slots. Since we have 6 managers, there will be 3 meetings going on concurrently in each time slot. The number of time slots required can be found by dividing the total number of pairings (15) by the number of simultaneous meetings (3). The minimum number of time slots required is \(15/3 = 5\). Since there are 3 meetings in each time slot, we can distribute the pairings as follows: _Time Slot 1_ AB, CD, EF _Time Slot 2_ AC, BD, EF _Time Slot 3_ AD, BE, CF _Time Slot 4_ AE, BF, CD _Time Slot 5_ AF, BC, DE

Step 3: Assign each pairing to the meeting time

Lastly, assign each pairing to a specific meeting time, using the 1-hour time slots and starting at 7 AM. _Time Slot 1: 7 AM_ - Meeting 1: A with B - Meeting 2: C with D - Meeting 3: E with F _Time Slot 2: 8 AM_ - Meeting 1: A with C - Meeting 2: B with D - Meeting 3: E with F _Time Slot 3: 9 AM_ - Meeting 1: A with D - Meeting 2: B with E - Meeting 3: C with F _Time Slot 4: 10 AM_ - Meeting 1: A with E - Meeting 2: B with F - Meeting 3: C with D _Time Slot 5: 11 AM_ - Meeting 1: A with F - Meeting 2: B with C - Meeting 3: D with E This schedule ensures that all 15 pairings have their meetings and that no manager has overlapping meetings.

Key Concepts

Combinatorics

Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of objects that satisfy certain criteria. It’s incredibly useful in creating schedules, like for a round-robin tournament, where each participant must meet every other participant.

In our example, the company has six managers needing to meet one another individually, akin to a mutual handshake scenario. To calculate the total number of handshakes, or pairings, we utilize a combinatorial formula known as the combination formula, represented as \( C(n, k) = \frac{n!}{k!(n-k)!} \), where 'n' is the total number of items, 'k' is the number to select, and '!' denotes a factorial. For pairwise meetings, 'k' is always 2 as we pick 2 managers at a time from the group of six, essentially calculating \( C(6, 2) \).

The solution gives us 15 unique pairings, highlighting the role of combinatorics in effectively organizing such complex scheduling requirements without redundancies or omissions.

Pairing Combinations

Pairing combinations involve grouping participants into unique pairs where each individual is matched with every other individual exactly once. This is pivotal in round-robin tournaments and similar scheduling situations.

In our example, each manager must meet with every other manager, resulting in a situation where we need to consider every possible unique pairing. To ensure clarity and prevent oversight, a list of all possible pairs is written down, which is what Step 1 in the solution entails and is a practical application of pairing combinations.

For educational purposes, understanding the generation of such pairings systematically helps ensure that all participants have the opportunity to interact with each other without repetition, which is especially important when each interaction (or meeting, in this case) carries significant value.

Time Slot Allocation

Time slot allocation involves organizing events or activities into specific periods, considered as slots, where each activity occurs without conflict. It's crucial for time management and resource optimization in scheduling multiple concurrent events.

In our scenario, allocating time slots begins after determining the total number of meetings. Because only three rooms are available (given three pairs can meet simultaneously), we divide the total number of pairings by three, leading to the necessity of five different time slots. This is concisely elucidated in Step 2 of the solution, ensuring that all meetings occur without overlap and within the confines of the working hours starting at 7 AM.

The concept highlights a real-world application of combinatorics and scheduling, providing learners with a relatable scenario that demonstrates the importance of structured time slot allocation. Understanding how to efficiently distribute activities maximizes resource usage and adherence to constraints, such as available time or spaces.