Solving the Mysterious Four-Digit Number Puzzle: A Mathematical Journey

A fascinating problem often catch the attention of math enthusiasts and puzzlers. Consider a four-digit number where the second digit is 0, and when the digits are reversed, the resulting number is exactly 9 times the original number. What is the original number? To crack this puzzle, we will approach it step by step, applying algebra and a process of elimination.

Understanding the Problem

Let's denote the four-digit number as (overline{a0bc}), where (a), (b), and (c) are the digits of the number. This can be expressed mathematically as:

[ N 1000a 0 cdot 100 10b c 1000a 10b c ]

Upon reversing the digits, the number becomes (overline{cb0a}), which can be expressed as:

[ R 1000c 0 cdot 100 10b a 1000c 10b a ]

According to the problem, the reversed number (R) is 9 times the original number (N):

[ 1000c 10b a 9(1000a 10b c) ]

Formulating and Solving the Equation

Expanding and simplifying the equation:

[ 1000c 10b a 9000a 90b 9c ] [ 1000c - 9c 10b - 90b a - 9000a 0 ] [ 991c - 8999a - 80b 0 ]

This equation can be rearranged to express a relationship among (a), (b), and (c):

[ 991c 8999a 80b ]

Finding Possible Values for (a), (b), and (c)

Solving for (c) and ensuring divisibility by 80, we check various values:

When (c 1): (991 cdot 1 991) (not divisible by 80) When (c 2): (991 cdot 2 1982) (not divisible by 80) When (c 3): (991 cdot 3 2973) (not divisible by 80) When (c 4): (991 cdot 4 3964) (not divisible by 80) When (c 5): (991 cdot 5 4955) (not divisible by 80) When (c 6): (991 cdot 6 5946) (not divisible by 80) When (c 7): (991 cdot 7 6937) (not divisible by 80) When (c 8): (991 cdot 8 7928) (not divisible by 80) When (c 9): (991 cdot 9 8919) (not divisible by 80)

It becomes clear that we need to find a combination where both sides are divisible by 991. We will test various values of (a), (b), and (c).

Testing Values

Testing (a 1):

[ 991c 8999 cdot 1 80b ]

Testing (c 9):

[ 991 cdot 9 8999 80b ] [ 8919 8999 80b ] [ 80b 8919 - 8999 -80 ]

This value of (b) is not valid. We continue this process until we find a valid combination.

Testing (a 2):

[ 991c 8999 cdot 2 80b ]

Testing (c 1):

[ 991 cdot 1 8998 80b ] [ 991 8998 80b ] [ 80b 991 - 8998 -8007 ]

Still not valid. We continue testing until we find the combination (a 1), (b 0), and (c 9):

[ 1000 cdot 1 10 cdot 0 9 1009 ]

Reversing the digits:

[ 9001 ]

We verify:

[ 9001 9 cdot 1009 ]

Thus, the original number is (boxed{1098}).