Hello, young mathematicians! Today, we have a fun and challenging math puzzle for you. This puzzle will help you understand the concept of ‘weight’ in numbers and how to use it to solve problems. So, let’s dive in!

Imagine you have nine different numbers: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Your task is to arrange these numbers into three different three-digit numbers. The catch is, you want the sum of these three numbers to be as large as possible. Sounds interesting, right?

First, let’s understand what ‘weight’ means in this context. The weight of a digit in a number is how much that digit contributes to the total value of the number. For example, in the number 123, the digit 1 has a weight of 100, the digit 2 has a weight of 10, and the digit 3 has a weight of 1.

Let’s denote the three numbers we want to create as $a_1a_2a_3$, $b_1b_2b_3$, and $c_1c_2c_3$. The sum of these three numbers can be written as:

$a_1a_2a_3 + b_1b_2b_3 + c_1c_2c_3$

This can be rearranged as:

$(a_1+b_1+c_1) * 100 + (a_2+b_2+c_2) * 10 + (a_3+b_3+c_3) * 1$

This shows us that the digits in the hundreds place contribute a hundred times more to the sum than the digits in the ones place. So, to maximize the sum, we should put the largest numbers in the hundreds place, the next largest in the tens place, and the smallest in the ones place.

Let’s fill in the weights and the numbers in a table:

digit $a_1$ $b_1$ $c_1$ $a_2$ $b_2$ $c_2$ $a_3$ $b_3$ $c_3$
weight 100 100 100 10 10 10 1 1 1
value 9 8 7 6 5 4 3 2 1

From this table, we can see that the three numbers we create are 963, 852, and 741. If we add these numbers together, we get a total of 2556. And that’s the largest sum we can get!

I hope you enjoyed this puzzle and learned something new about the weight of digits in a number. Keep practicing and stay curious!