Deciphering the Mystery of the Digit-less Number

Have you ever come across a challenge that combines number theory with a bit of wordplay? Let's explore a fascinating riddle involving a three-digit number, where each digit interrelates in a peculiar manner. This puzzle will not only test your analytical skills but also entertain your curiosity in the realm of mathematics.

The Problem Statement

Find a three-digit number where:

The second digit is 4 times the third digit. The first digit is 2 less than the second digit.

Solving the Puzzle Step-by-Step

To solve this intriguing problem, let's denote the three-digit number as ABC, where A is the first digit, B is the second digit, and C is the third digit. Now, let's break down the clues systematically.

Clue 1: The Second Digit is 4 Times the Third Digit

This clue translates to a simple equation: B 4C. Since B and C are digits (0-9), let's explore the possible values for C and their corresponding B values:

If C 0, then B 4 * 0 0. In this case, A B - 2 0 - 2 -2, which is not a valid digit (since digits must be between 0 and 9).

If C 1, then B 4 * 1 4. In this case, A B - 2 4 - 2 2. This gives us the number 241, which is a valid three-digit number.

If C 2, then B 4 * 2 8. In this case, A B - 2 8 - 2 6. This gives us the number 682, which is also a valid three-digit number.

Conclusion: Valid Three-Digit Numbers

Thus, the three-digit numbers that satisfy the given conditions are 241 and 682. These are the only possible solutions to our problem.

Further Exploration

Let's see if we can derive the same result using a different approach. Consider a three-digit number (xyz), where:

y 4z x y - 3

Substituting the value of y from the second equation into the first, we get:

x 4z - 3

Since x, y, and z must be digits (0-9), let's evaluate the possible values for z, and hence x and y using a hit-and-trial method:

For z 0, x 4(0) - 3 -3, y 0. This does not yield a valid three-digit number since digits cannot be negative.

For z 1, x 4(1) - 3 1, y 4. This yields the number 141, which is a valid three-digit number.

For z 2, x 4(2) - 3 5, y 8. This yields the number 582, which is also a valid three-digit number.

As we can see, the calculations and the conclusions match with the previous steps, reinforcing our findings.

Final Answer

Therefore, the valid three-digit numbers that satisfy the conditions are 241 and 582.