Key Concepts: Weight, Minima

Let’s dive into a fun math problem today! Imagine you have the numbers 1, 3, 5, 7, 9 and you need to fill them into the five boxes on the left side of an equation. On the right side, you have the numbers 0, 2, 4, 6, 8 to fill into the five boxes. The equation looks like this:

□÷□+□+□□=□+□+□□÷□

The challenge is to make the equation true, with both sides resulting in natural numbers. The question is, what is the smallest possible result?

First, let’s look at the left side of the equation. We only have addition and division operations. The smallest result will be achieved when the divisor is the largest. To ensure that □÷□ does not leave a remainder, the divisor can only be 1 or 3. Clearly, when the divisor is 3, the result is the smallest. Therefore, the dividend must be 9.

Now, we need to fill the remaining boxes with 1, 5, and 7 to get the smallest possible result. Let’s assign these numbers to variables a, b, and c. We can create a weight table for these numbers:

Number a b c
Weight 1 10 1

By sorting the numbers according to their weights, we get:

Number a c b
Weight 1 1 10
Value 7 5 1

So, the left side of the equation becomes □÷3+□+□□ = 9/3 + 7 + 15 = 25.

Now, let’s move to the right side of the equation. We need to make □+□+□□÷□ = 25. Since the first two numbers are even, the last number must be odd to ensure the result is 25. To keep the result small, the divisor should be as large as possible.

After some trial and error, we find that the divisor cannot be 8 or 6. When the divisor is 4, the dividend can be 20, 28, or 60 to get an odd result. After testing these possibilities, we find that □□÷□ = 60/4 makes the equation true.

So, the smallest possible result for □÷□+□+□□=□+□+□□÷□ is 25, with the numbers filled in as follows: 9/3 + 7 + 15 = 2 + 8 + 60/4.

This problem is a great example of how understanding the properties of numbers and operations can help us solve complex problems. It’s like a puzzle, where each piece has to fit just right. And that’s what makes math so much fun!