Solve The Problem By Explaining The Solution. Kolya Thought Of A Two-digit Number, Increased It Threefold, Reduced The Result By Two, And Obtained A Number Written With The Same Digits As The Original, But In A Different Order. What Number Did Kolya Think Of?

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Have you ever encountered a mathematical riddle that seems simple on the surface but requires careful thought to unravel? This is precisely the case with the puzzle Kolya has presented. Let's dive into this intriguing problem, break down the steps, and discover the solution together. This article will explore the solution to a fascinating mathematical puzzle, focusing on a step-by-step approach to understanding the logic and arriving at the correct answer. We will dissect the problem, identify the key elements, and then apply a systematic method to decode the mystery of Kolya's two-digit number. This exploration aims to not only provide the solution but also to enhance your problem-solving skills and logical reasoning.

Unraveling the Enigma of Kolya's Number

Kolya's puzzle presents us with a fascinating challenge: a two-digit number is tripled, then decreased by two, resulting in a number formed by the same digits but in a different order. The core of this problem lies in understanding number properties and applying logical deduction. We are not just looking for an answer; we are embarking on a journey of mathematical reasoning. The beauty of such problems is that they encourage us to think creatively and systematically, skills that are invaluable in various aspects of life.

Defining the Problem Statement

The problem statement can be clearly defined as follows: Kolya is thinking of a two-digit number. He multiplies this number by 3, subtracts 2 from the result, and obtains a new two-digit number. This new number is formed using the same digits as the original number but in a reversed order. The task is to determine the original two-digit number that Kolya had in mind. This seemingly simple puzzle involves understanding the properties of two-digit numbers, basic arithmetic operations, and logical deduction. The challenge is to find the number that satisfies all the given conditions, making it a compelling exercise in mathematical problem-solving.

Laying the Foundation: Algebraic Representation

To solve this puzzle efficiently, we first need to represent the problem algebraically. Let's denote the two-digit number Kolya initially thought of as "10a + b," where "a" represents the tens digit and "b" represents the units digit. Both "a" and "b" are integers between 0 and 9, with "a" not being zero since it's a two-digit number. The new number, formed by reversing the digits, can be represented as "10b + a." Now, we can translate the problem's conditions into an algebraic equation. According to the problem, three times the original number, minus two, equals the reversed number. This gives us the equation: 3(10a + b) - 2 = 10b + a. This equation is the cornerstone of our solution, allowing us to manipulate the variables and find the values of "a" and "b" that satisfy the conditions. By expressing the problem in this algebraic form, we have laid a solid foundation for further analysis and solution.

The Equation Unveiled: 3(10a + b) - 2 = 10b + a

The equation 3(10a + b) - 2 = 10b + a is the mathematical heart of our problem. It encapsulates the relationship between the original two-digit number and the number formed by reversing its digits, after Kolya's manipulations. To solve this, we need to simplify and rearrange the equation to a more manageable form. Expanding the left side, we get 30a + 3b - 2 = 10b + a. Now, let's group the terms with 'a' and 'b' on one side and the constant term on the other. Subtracting 'a' from both sides and subtracting '3b' from both sides, we get 29a - 7b = 2. This simplified equation is crucial because it reduces the complexity of the problem. It allows us to focus on the relationship between 'a' and 'b' and to systematically test possible values to find a solution. The ability to transform an initial equation into a simpler form is a fundamental skill in mathematical problem-solving, and this step demonstrates its importance in deciphering Kolya's numerical riddle.

Cracking the Code: Solving the Equation 29a - 7b = 2

Now, we arrive at the critical stage of solving the equation 29a - 7b = 2. This equation is a Diophantine equation, which means we are looking for integer solutions for 'a' and 'b'. Recall that 'a' and 'b' are digits, so they must be integers between 0 and 9, with 'a' being non-zero. The challenge here is to find the specific pair of digits that satisfy this equation. We can approach this in several ways, such as trial and error, substitution, or modular arithmetic. However, a systematic approach is often the most efficient. One method is to isolate one variable and express it in terms of the other. For instance, we can rewrite the equation as 29a = 7b + 2. From this, we can express 'a' as a = (7b + 2) / 29. Now, we can test different integer values of 'b' (from 0 to 9) and see which one results in an integer value for 'a' that also falls between 1 and 9. This process might seem tedious, but it's a direct and reliable way to find the solution. By systematically exploring the possible values of 'b', we can identify the pair of digits that fit the equation and unlock the mystery of Kolya's number.

Methodical Deduction: Finding the Right Digits

To solve the equation 29a - 7b = 2, we employ a methodical deduction strategy. We know that 'a' and 'b' are single-digit integers, and 'a' cannot be zero. Our approach is to test values for 'b' from 0 to 9 and check if the resulting 'a' is also a single-digit integer. This method ensures we systematically explore all possibilities. Let's begin by rearranging the equation to isolate 'a': a = (7b + 2) / 29. Now, we substitute different values for 'b' and see if we get an integer value for 'a'.

  • If b = 0, then a = (7(0) + 2) / 29 = 2 / 29 (not an integer)
  • If b = 1, then a = (7(1) + 2) / 29 = 9 / 29 (not an integer)
  • If b = 2, then a = (7(2) + 2) / 29 = 16 / 29 (not an integer)
  • If b = 3, then a = (7(3) + 2) / 29 = 23 / 29 (not an integer)
  • If b = 4, then a = (7(4) + 2) / 29 = 30 / 29 (not an integer)
  • If b = 5, then a = (7(5) + 2) / 29 = 37 / 29 (not an integer)
  • If b = 6, then a = (7(6) + 2) / 29 = 44 / 29 (not an integer)
  • If b = 7, then a = (7(7) + 2) / 29 = 51 / 29 (not an integer)
  • If b = 8, then a = (7(8) + 2) / 29 = 58 / 29 (not an integer)
  • If b = 9, then a = (7(9) + 2) / 29 = 65 / 29 (not an integer)

It seems like our initial approach didn't yield an immediate solution. Let's rethink our strategy. Instead of isolating 'a', let's try isolating 'b': b = (29a - 2) / 7. Now we'll test values for 'a' from 1 to 9.

  • If a = 1, then b = (29(1) - 2) / 7 = 27 / 7 (not an integer)
  • If a = 2, then b = (29(2) - 2) / 7 = 56 / 7 = 8 (an integer!)

We found a solution! When a = 2, b = 8. This methodical approach demonstrates the power of systematic testing in solving Diophantine equations. By carefully considering the constraints of the problem and applying a step-by-step method, we have successfully identified the digits that satisfy the equation. This breakthrough brings us closer to uncovering Kolya's mysterious number.

The Solution Emerges: a = 2, b = 8

Through our methodical deduction, we've discovered that a = 2 and b = 8 satisfy the equation 29a - 7b = 2. This is a significant milestone in solving the puzzle. These values represent the digits of Kolya's original number. Remember, 'a' is the tens digit and 'b' is the units digit. Now that we have these values, we can construct the original number and verify our solution. The excitement of arriving at a solution after a logical process is one of the joys of mathematical problem-solving. This discovery not only provides us with the answer but also reinforces the effectiveness of a systematic approach in tackling complex problems.

Putting the Pieces Together: Constructing Kolya's Number

With a = 2 and b = 8, we can now reconstruct Kolya's original two-digit number. Recall that we represented the number as 10a + b. Substituting the values we found, the original number is 10(2) + 8 = 20 + 8 = 28. So, Kolya initially thought of the number 28. But our journey isn't over yet. We need to verify that this number satisfies all the conditions of the problem. This step is crucial to ensure the accuracy of our solution and to complete the puzzle. It's a testament to the importance of thoroughness in problem-solving, where verifying the answer is as crucial as finding it.

Verifying the Solution: The Final Check

To ensure our solution is correct, we need to verify that the number 28 satisfies the conditions of the problem. The conditions were: (1) Multiply the original number by 3, (2) Subtract 2 from the result, and (3) The final number should be the original digits reversed. Let's perform these operations on 28.

  1. Multiply by 3: 28 * 3 = 84
  2. Subtract 2: 84 - 2 = 82

The resulting number is 82. Now, let's check if the digits of 82 are the digits of 28 reversed. Indeed, they are! The digits 2 and 8 have been reversed to form 82. This final check confirms that our solution is correct. It demonstrates the importance of not only finding an answer but also verifying that it fits all the criteria of the problem. The satisfaction of solving a puzzle completely, with all loose ends tied up, is a rewarding experience.

Conclusion: Kolya's Number Revealed as 28

In conclusion, after a step-by-step journey through the problem, we have successfully decoded Kolya's two-digit mystery. By translating the problem into an algebraic equation, systematically solving for the digits, and verifying our solution, we have discovered that the number Kolya initially thought of was 28. This exercise exemplifies the power of mathematical reasoning and the importance of a structured approach to problem-solving. It highlights how seemingly complex puzzles can be unraveled with careful analysis, logical deduction, and a touch of perseverance. The satisfaction of solving such a puzzle lies not just in finding the answer but also in the intellectual journey and the skills honed along the way. This experience encourages us to approach future challenges with confidence and a methodical mindset.

By following the steps outlined in this article, you can apply these problem-solving techniques to similar mathematical puzzles and enhance your analytical abilities. The world of mathematics is full of such fascinating challenges, and with the right tools and approach, they can be both enjoyable and rewarding to solve. So, keep exploring, keep questioning, and keep solving!