Dividing Coffee Mugs Equally: A Mathematical Puzzler

Imagine you are in a room with 14 coffee mugs arranged on a table, where 7 are filled with coffee and 7 are half-filled. The challenge is to divide these mugs into three groups, ensuring that each group has the same total amount of coffee. This seemingly simple task involves a bit of mathematical creativity. Let's dive into the solution and explore the underlying principles.

Understanding the Challenge

First, let's break down the situation: you have 7 full mugs and 7 half-full mugs, resulting in a total volume of 73.5 cups of coffee. When you divide 73.5 by 3 groups, you get 24.5 cups per group. However, since we can't change the amount of coffee in any mug, our goal is to distribute the mugs themselves in a way that achieves the same total amount of coffee in each group.

The Solution

One effective approach is to distribute the mugs in a way that each group contains a combination of full and half-full cups. Here's how we can do it:

Group 1:

3 full cups (3 cups) 1 half cup (0.5 cups) 3.5 cups in total

Group 2:

3 full cups (3 cups) 1 half cup (0.5 cups) 3.5 cups in total

Group 3:

1 full cup (1 cup) 5 half cups (2.5 cups) 3.5 cups in total

Therefore, each group contains a total of 3.5 cups of coffee.

Another Approach

Another way to approach this problem is to start by dividing the 14 physical cups into 3 groups. There are 7 full cups and 7 half-cups, so we need to distribute them equally. Here's how you can do it:

Group 1:

2 full cups (2 cups) 3 half cups (1.5 cups) 3.5 cups in total

Group 2:

2 full cups (2 cups) 3 half cups (1.5 cups) 3.5 cups in total

Group 3:

3 full cups (3 cups) 1 half cup (0.5 cups) 3.5 cups in total

Again, each group ends up with a total of 3.5 cups of coffee, ensuring that the distribution is fair.

The Mathematical Explanation

From a mathematical standpoint, we can approach the problem by recognizing that the total volume of coffee is 10.5 cups (7 full cups 3.5 cups from the half-cups). Dividing 10.5 by 3 gives us 3.5 cups per group.

Breakdown:

- 7 full cups: 7 * 1.5 10.5 cups

- 7 half cups: 7 * 0.5 3.5 cups

- Total: 10.5 cups

- Dividing by 3: 10.5 / 3 3.5 cups per group

Conclusion

By carefully considering the combination of full and half-full cups, you can divide the 14 mugs into three groups, each containing a total of 3.5 cups of coffee. This puzzle demonstrates the power of logical reasoning and the importance of understanding the underlying principles of distribution.

Now that you've solved this intriguing puzzle, you might want to explore more mathematical challenges or even use this knowledge to optimize other distribution scenarios in your daily life.