Hello, young mathematicians! Today, we’re going to play a fun game with numbers. We’ll use the numbers 0 to 9 to create two five-digit numbers. Our goal is to find the combinations that give us the largest and smallest products. Sounds exciting, right? Let’s dive in!

Before we dive into the problem, let’s learn a new concept: weight. The weight of a digit in a number is its power to influence the final value of the number. If we want the final number to be the largest, we should assign larger digits to positions with higher weight.

For example, in the number abc, composed of digits 1, 2, and 3, abc = a * 100 + b * 10 + c * 1. Here, ‘a’ has the highest weight (100), ‘b’ has the next highest weight (10), and ‘c’ has the lowest weight (1). To make this number the largest, we should put the largest digit in the position with the highest weight. So, 321 is the largest number we can form, and 123 is the smallest. Now, let’s get back to our problem. Let’s assume the two numbers we form are abcde and hijkl, with abcde >= hijkl. Their multiplication can be approximated as: \(abcde * hijkl ≈ (a*h)*100000000 + (b*i)*1000000 + (c*j)*10000 + (d*k)*100 + (e*l)\)

To maximize this multiplication, we should assign the digits as follows: $(a,h) = (9,8) , (b|i) = (7|6), (c|j) = (5|4), (d|k) = (3|2), (e|l) = (1|0)$

Here, $(a,h) = (9,8)$ means $a=9$ and $h=8$. $(b|i) = (7|6)$ means $b=7$ and $i=6$, or $b=6$ and $i=7$. But which one gives a larger multiplication, $(b,i) = (7,6)$ or $(b,i) = (6,7)$?

We know that the sum of ab and hi is the same in both cases. But when the sum of two numbers is fixed, their multiplication is larger when the difference between them is smaller. So, we should assign the larger digit, 7, to the smaller number, hijkl. Therefore, we have $(b,i) = (6,7)$, and similarly,$(c,j)=(4,5),(d,k)=(2,3),(e,l)=(0,1)$.

So, the two numbers that give the largest multiplication are $96420$ and $87531$, and their multiplication is $96420*87531 = 8439739020$.

Using the same logic, we can find the two numbers that give the smallest multiplication. But there’s a special case here: the highest digit cannot be 0, because then the number would become a four-digit number instead of a five-digit number.

So, we should assign the digits as follows: $(a,h) = (2,1), (b|i) = (0|3), (c|j) = (4|5), (d|k) = (6|7), (e|l) = (8|9)$

Again, when the sum of two numbers is fixed, their multiplication is smaller when the difference between them is larger. So, we have $(a,h) = (2,1), (b,i) = (3,0), (c,j) = (5,4), (d,k) = (7,6), (e,l) = (9,8)$.

So, the two numbers that give the smallest multiplication are $21057$ and $398644$, and their multiplication is $21057*39864 = 839485728$.

Isn’t it amazing how the arrangement of digits can drastically change the multiplication of two numbers? Keep exploring the magic of numbers, and you’ll discover more fascinating facts!