Solving Three-Digit Number Puzzles: A Mathematical Journey

Solving Three-Digit Number Puzzles: A Mathematical Journey

Introduction to Three-Digit Number Puzzles

Three-digit number puzzles are a fascinating area of mathematical recreation. These puzzles often involve logical reasoning and algebraic manipulation to uncover hidden numerical relationships. In this article, we will explore a specific problem that challenges us to find a three-digit number with certain conditions. We will employ algebraic techniques to solve the problem step-by-step and verify the solution. This approach not only helps in understanding the problem but also enhances logical thinking and problem-solving skills.

Understanding the Problem

We are given a three-digit number (abc), where (a), (b), and (c) are the digits of the number. The conditions to be satisfied are as follows: The sum of the digits is 10. The middle digit (b) is equal to the sum of the other two digits (a) and (c). The number increases by 99 when its digits are reversed. We need to find the three-digit number that meets all these conditions.

Step-by-Step Solution

Let's denote the three-digit number as (abc), where (a), (b), and (c) are the digits. We can write the number as 100a 10b c. The conditions can be expressed algebraically as follows:

Condition 1: Sum of the Digits

Equation 1: [a b c 10]

Condition 2: Middle Digit as Sum of Other Two

Equation 2: [b a c]

Condition 3: Increase by 99 When Reversed

Equation 3: [100a 10b c 99 100c 10b a]

Simplifying the Third Equation

Starting with the third equation, we simplify it as follows: 100a 10b c 99 100c 10b a Subtracting (10b) from both sides gives: 100a c 99 100c a Rearranging the terms: 100a - a c - 100c -99 99a - 99c -99 Dividing the entire equation by 99: a - c -1 Thus, we can express (c) in terms of (a): Equation 4: [c a 1]

Substituting (c) in the First and Second Equations

Now, we substitute (c a 1) into both the first and second equations. From the first equation (a b c 10): a b (a 1) 10 Simplifying: 2a b 1 10 2a b 9 quad text{(Equation 1)} From the second equation (b a c): b a (a 1) 2a 1 Equation 2: [b 2a 1]

Solving the Equations

Substitute Equation 2 into Equation 1: 2a (2a 1) 9 Simplifying: 4a 1 9 Subtracting 1 from both sides: 4a 8 Dividing by 4: a 2 Now, substitute (a 2) back into the equations for (b) and (c). From Equation 2: b 2(2) 1 5 From Equation 4: b 2 1 3 Therefore, we have: a 2, b 5, c 3 The three-digit number is (253).

Verification

Let's verify the solution by checking the given conditions: The sum of the digits: 2 5 3 10. The middle digit equals the sum of the other two: 5 2 3. The number increases by 99 when its digits are reversed: 352 and 253 99 352. All conditions are satisfied, confirming that the number is indeed 253.

Alternative Example

For an alternative example, let's solve a similar puzzle where (abc 14). This example will illustrate a different approach and verify the solution:

abc 14 implies b 14 - a - c. b ac implies 2a^2c 14 implies b 7 and c 7 - a. 100c - 10b - a 100a - 10b - c - 99. implies 99c - 99a 99. implies c - a 1. implies 7 - a - a 1. implies 2a 6. boxed{a, b, c 3, 7, 4}, The number is 374.

Conclusion

We have successfully solved the three-digit number puzzle and verified the solution using algebraic reasoning. This method not only solves the problem but also provides a clear understanding of the underlying mathematical principles. Such puzzles are excellent for enhancing logical and problem-solving skills, making them a valuable exercise for anyone interested in mathematics or puzzle-solving.