Six-Digit Numbers: Exactly One Pair Differs By 5

Aug 3, 2025 by ADMIN 49 views

Six-Digit Numbers with a Digit Difference of 5: A Comprehensive Guide

Hey guys! Ever wondered about the fascinating world of numbers and their quirky properties? Today, we're diving deep into a combinatorics problem that involves six-digit numbers and a specific condition: exactly one pair of adjacent digits must differ by 5. Let's break down the problem, explore the solution, and uncover the magic behind these numerical puzzles. This comprehensive guide will not only provide the answer but also equip you with the skills to tackle similar problems with confidence.

Understanding the Problem: The Six-Digit Number Puzzle

In this intriguing number puzzle, we're tasked with finding the count of six-digit numbers that adhere to a unique constraint. The core of the problem lies in the phrase "exactly one pair of adjacent digits that differ by 5." This seemingly simple condition opens up a world of possibilities, but also demands a systematic approach to avoid overcounting or missing potential candidates. To truly grasp the challenge, let's dissect the key components:

Six-Digit Numbers: We're dealing with numbers ranging from 100,000 to 999,999. The first digit cannot be zero, adding a slight twist to the calculations.
Adjacent Digits: This refers to digits that are next to each other in the number (e.g., in 794880, 7 and 9 are adjacent, 9 and 4 are adjacent, and so on).
Differ by 5: The absolute difference between two adjacent digits must be exactly 5. For instance, the pairs (0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 0), (6, 1), (7, 2), (8, 3), and (9, 4) satisfy this condition.
Exactly One Pair: This is the most crucial constraint. Only one pair of adjacent digits within the six-digit number can have a difference of 5. This means all other adjacent digit pairs must have differences other than 5.

The number 794880 serves as an excellent example. The adjacent digits 4 and 9 have a difference of 5, and this is the only such pair in the number. Numbers like 161111 (two pairs: 1 and 6) or 123456 (no pairs) would not fit the criteria. So, our mission is to count how many numbers, in total, meet this specific requirement. This requires a methodical approach, combining combinatorial principles with careful case analysis.

Breaking Down the Solution: A Step-by-Step Approach

To solve this intricate problem, we need a strategy that ensures accuracy and avoids overlooking any valid numbers. Here's a breakdown of the step-by-step approach we'll use:

Identify Possible Pairs: First, let's list all possible pairs of digits that have a difference of 5. As we discussed earlier, these pairs are (0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 0), (6, 1), (7, 2), (8, 3), and (9, 4). These are the building blocks for our numbers.
Determine Positions for the Pair: Since we need exactly one pair of adjacent digits differing by 5, we need to figure out where this pair can be positioned within the six-digit number. There are five possible positions: (1st and 2nd digits), (2nd and 3rd digits), (3rd and 4th digits), (4th and 5th digits), and (5th and 6th digits).
Consider Cases Based on the Pair's Position: For each position of the special pair, we need to consider all possible digit combinations for the remaining slots. This is where things get interesting, as we must ensure that no other adjacent digits have a difference of 5. This constraint significantly reduces the number of valid options.
Calculate Valid Digit Combinations: For each position of the special pair, we need to carefully calculate the number of valid digit combinations for the remaining four digits. This involves considering the restrictions imposed by the “exactly one pair” condition. We'll need to think about what digits are not allowed in certain positions to avoid creating another pair with a difference of 5.
Sum the Results from All Cases: After calculating the possibilities for each position of the pair, we sum up the results to get the total number of six-digit numbers that satisfy the given condition. This final summation will give us our answer.
Careful Consideration of the First Digit: Remember that the first digit of a six-digit number cannot be zero. This restriction needs to be accounted for, especially when the pair differing by 5 involves a 0.
Eliminating Overcounting: It's crucial to avoid overcounting. Double-check that each case is mutually exclusive and that no number is counted more than once. This is where careful logic and organization are paramount.

By systematically following these steps, we can break down the problem into manageable chunks and arrive at the correct solution. Let’s start applying this strategy in the next section!

Case-by-Case Analysis: Position Matters!

Now, let's put our strategy into action by analyzing each possible position for the digit pair that differs by 5. This case-by-case approach will help us systematically count the numbers while adhering to the "exactly one pair" rule. We’ll go through each position, considering the constraints and calculating the valid combinations.

Case 1: The Pair is in the 1st and 2nd Positions

In this scenario, the first two digits have a difference of 5. This means we have pairs like 16, 27, 38, 49, 50, 61, 72, 83, 94, and 50 as the first two digits of our six-digit number. Now, let's delve into the implications for the remaining four digits:

First Digit Cannot Be Zero: This is a fundamental rule for six-digit numbers, so pairs like 05 are not valid in this position. This means we start with 9 possible pairs.
Constraints on the 3rd Digit: The third digit cannot form a pair with the second digit that also has a difference of 5. For each choice of the first two digits, we need to eliminate the digit that creates a difference of 5 with the second digit. For instance, if the first two digits are 16, the third digit cannot be 1 (because 6-1 = 5). This leaves us with 9 possibilities for the third digit (0-9, excluding 1).
Constraints on Subsequent Digits: Similarly, the fourth digit cannot have a difference of 5 with the third, the fifth cannot differ by 5 from the fourth, and the sixth cannot differ by 5 from the fifth. For each of these digits, we'll have to subtract the disallowed digit, typically leaving us with 9 possibilities. This part is crucial, guys, as it embodies the “exactly one pair” condition.

To calculate the total numbers in this case, we multiply the possibilities for each digit. However, there is a small caveat: if the first two digits are 50, we have to account for the restriction that the first digit cannot be 0. So, we'll handle this case slightly differently. We'll calculate the possibilities for other pairs first and then add the possibilities for the 50 pair.

Case 2: The Pair is in the 2nd and 3rd Positions

Now, let's shift our focus to when the pair with a difference of 5 is in the second and third positions. This introduces a new set of considerations, particularly for the first digit:

First Digit (Again) Cannot Be Zero: Remember, this rule remains in effect. The first digit can be any number from 1 to 9.
Constraints on the First Digit: The first digit cannot have a difference of 5 with the second digit. This is critical. If the second digit is, say, 6, then the first digit cannot be 1. This constraint further limits our possibilities.
The Pair in the 2nd and 3rd Position: We still have the same 10 pairs (05, 16, 27, 38, 49, 50, 61, 72, 83, 94) to consider for these positions. These pairs are our focus in this case.
Constraints on the Remaining Digits: The fourth digit cannot differ by 5 from the third, the fifth cannot differ by 5 from the fourth, and the sixth cannot differ by 5 from the fifth. This “exactly one pair” rule continues to be our guiding principle.

The calculations for this case are slightly more complex due to the interplay between the first and second digits. We'll need to carefully consider each pair in the second and third positions and determine the allowable digits for the first position accordingly. This might seem tricky, but trust me, breaking it down step-by-step makes it manageable.

Cases 3, 4, and 5: Pair in the 3rd & 4th, 4th & 5th, and 5th & 6th Positions

The logic for Cases 3, 4, and 5 is similar to Case 2, with the pair occupying different positions. The key is to recognize that the constraints shift depending on the pair's location. For example, when the pair is in the 5th and 6th positions, the first four digits are subject to the “no adjacent difference of 5” rule, while the last two digits are fixed as one of our ten pairs.

For each case, we’ll need to:

Consider the First Digit Constraint: Always remember that the first digit cannot be zero.
Account for “No Adjacent Difference of 5”: This rule applies to all adjacent digits except for the designated pair.
Multiply the Possibilities: After figuring out the allowable digits for each position, we multiply the numbers to find the total combinations for that case.

By systematically working through each case, we are ensuring that we capture every possible six-digit number that meets our criteria. It’s like detective work, you know? We're piecing together the clues to solve the puzzle!

Putting It All Together: The Final Calculation

After meticulously analyzing each case, we arrive at the final stage: summing up the results to obtain the total number of six-digit numbers that meet our criteria. This final calculation requires careful bookkeeping and attention to detail. Let's recap what we've done so far:

Identified the Possible Pairs: We listed all ten pairs of digits that have a difference of 5.
Determined the Positions: We considered all five possible positions for the special pair within the six-digit number.
Analyzed Each Case: We carefully examined each case, considering the constraints imposed by the “exactly one pair” rule and the “first digit cannot be zero” rule.
Calculated Valid Combinations: For each case, we calculated the number of valid digit combinations for the remaining slots.

Now, it's time to sum the valid combinations from each case. The individual case results will likely vary depending on the position of the pair and the associated constraints. For instance, Case 1 (pair in the 1st and 2nd positions) might yield a different number of valid numbers than Case 5 (pair in the 5th and 6th positions). It’s like adding up the pieces of a puzzle; each piece (case) contributes to the final picture (the total count).

Summing these individual totals will give us the grand total – the answer to our initial question. This number represents the total count of six-digit numbers that have exactly one pair of adjacent digits differing by 5. The process might seem long and complex, but the systematic approach ensures accuracy and comprehensiveness. It’s a testament to how combinatorics problems can be broken down into manageable steps, don't you think?

Tips and Tricks for Similar Problems

Now that we've conquered this specific problem, let's talk about some broader strategies and tricks that can help you tackle similar combinatorics challenges in the future. Problem-solving skills are like muscles – the more you exercise them, the stronger they become. So, let’s arm ourselves with some valuable techniques!

Break It Down: The most crucial strategy is to break down complex problems into smaller, more manageable parts. This is what we did with the six-digit number problem, analyzing each case separately.
Identify Key Constraints: Pay close attention to the constraints given in the problem. These constraints often act as guiding principles, helping you narrow down possibilities and avoid overcounting. In our problem, the “exactly one pair” rule was the key constraint.
Use Case Analysis: Case analysis is a powerful tool for tackling many combinatorics problems. By dividing the problem into distinct cases, you can systematically explore all possibilities without missing any or double-counting.
Consider Complementary Counting: Sometimes, it's easier to count the number of cases that don't satisfy the condition and subtract that from the total number of possibilities. This is known as complementary counting, and it can be a clever shortcut in certain situations.
Look for Patterns: Many combinatorics problems involve patterns. Identifying these patterns can simplify the counting process and lead to elegant solutions. For example, you might notice a recursive relationship or a symmetry in the problem.
Practice, Practice, Practice: Like any skill, problem-solving improves with practice. The more problems you solve, the better you'll become at recognizing common patterns and applying effective strategies.

By mastering these tips and tricks, you'll be well-equipped to tackle a wide range of combinatorics problems. Remember, the key is to approach each problem with a systematic mindset, break it down into manageable steps, and carefully consider all constraints. And, most importantly, have fun with it!

Conclusion: The Beauty of Combinatorics

So, we've journeyed through the world of six-digit numbers, explored the intricacies of adjacent digit differences, and emerged with a solution to our numerical puzzle. We've learned that even seemingly complex problems can be tackled with a methodical approach, careful consideration of constraints, and a dash of creativity. This problem, like many in combinatorics, showcases the beauty of mathematical problem-solving. It's not just about finding the right answer; it's about the journey of discovery, the development of logical thinking, and the satisfaction of cracking a challenging puzzle. I hope this comprehensive guide has been both informative and inspiring, and that it empowers you to tackle future combinatorics challenges with confidence. Keep exploring the fascinating world of numbers, my friends, and remember that every problem is an opportunity to learn and grow!