Solve Integer Puzzle Find The Number

Let's embark on a mathematical journey to unravel this integer puzzle! In this article, we'll dive deep into the realm of numbers, exploring the fascinating relationships between integers, their multiples, and the differences that arise between them. Prepare to engage your analytical prowess as we dissect the problem, develop a strategic approach, and arrive at the solution. This is not just about finding an answer; it's about understanding the underlying principles of number theory and honing our problem-solving skills.

Understanding the Core Concepts

Before we delve into the specifics of the puzzle, let's solidify our understanding of the fundamental concepts involved. Integers, the building blocks of our number system, encompass whole numbers, both positive and negative, as well as zero. The concept of "the next integer" is straightforward – it's simply the integer that follows the given one in the number line sequence. For instance, the next integer after 5 is 6, and the next integer after -3 is -2.

Multiplication plays a crucial role in this puzzle, specifically the concept of triples. A triple of a number is simply the result of multiplying that number by 3. The triple of 7 is 21, and the triple of -4 is -12. Finally, the notion of the "difference" between two numbers refers to the result of subtracting one number from the other. For example, the difference between 10 and 4 is 6 (10 - 4 = 6).

With these core concepts firmly in place, we're well-equipped to tackle the integer puzzle at hand. Remember, the key to success in mathematics lies not just in memorizing formulas, but in truly understanding the underlying principles. Let's proceed to dissect the puzzle statement and translate it into a mathematical equation.

Keywords Involved in the Integer Puzzle

To effectively solve any mathematical puzzle, it's essential to identify the keywords that hold the key to unlocking the solution. In this particular puzzle, several keywords stand out, each playing a crucial role in our understanding and approach. Let's break down these keywords and their significance:

• Integer: This word establishes the type of number we're dealing with. It tells us that the solution must be a whole number, without any fractional or decimal parts. This constraint helps narrow down the possible answers.
• Difference: This keyword signals the operation of subtraction. We know that we'll need to find the difference between two quantities, indicating that we'll be subtracting one from the other.
• Triple: This word indicates multiplication by 3. We'll encounter the need to find the triple of certain integers, meaning we'll multiply those integers by 3.
• Next Integer: This phrase specifies the integer that immediately follows a given integer in the number line sequence. Understanding this concept is crucial for correctly interpreting the puzzle statement.

By carefully analyzing these keywords, we can begin to form a mental model of the puzzle and identify the steps required to solve it. We'll use these keywords as our guide as we translate the puzzle into a mathematical equation.

Translating the Puzzle into an Equation

Now comes the exciting part: translating the verbal puzzle into a mathematical equation. This is where we transform words into symbols and establish the relationships between the quantities involved. Let's break down the puzzle statement piece by piece and convert it into its algebraic equivalent.

The puzzle asks us to find an integer that is equal to the difference between two quantities. Let's represent this unknown integer with the variable x. So, our equation will start with: x = ...

The first quantity we encounter is "the triple of the next integer." Let's say the integer we're looking for is n. Then, the next integer would be n + 1. The triple of the next integer is therefore 3(n + 1).

The second quantity is "the triple of the number 16," which is simply 3 * 16. The difference between these two quantities is what our unknown integer x is equal to. Putting it all together, we get the equation:

x = 3(n + 1) - 3 * 16*

This equation now represents the heart of the puzzle. It expresses the relationship between the unknown integer x and another integer n. Our next step is to simplify this equation and explore ways to solve for x. Remember, the beauty of algebra lies in its ability to capture complex relationships in concise symbolic form.

Solving the Equation Step by Step

With our equation in hand, let's embark on the journey of solving it step by step. This process involves applying algebraic principles to simplify the equation and isolate the unknown variable. We'll break down each step, explaining the reasoning behind it and ensuring clarity in our approach.

Our equation, as we derived earlier, is:

x = 3(n + 1) - 3 * 16*

The first step is to simplify the equation by performing the multiplication operations. Let's distribute the 3 in the first term:

x = 3n + 3 - 3 * 16***

Now, let's calculate 3 * 16, which equals 48:

x = 3n + 3 - 48***

Next, we can combine the constant terms (3 and -48):

x = 3n - 45***

At this point, we have a simplified equation that expresses x in terms of n. However, we still have two unknowns (x and n) and only one equation. This means we have a Diophantine equation, which may have multiple integer solutions. The problem statement implies there is a unique solution for x, so we need to use the information that x is the integer that is equal to the difference, implying x = n. Thus we can substitute x for n in the equation:

x = 3x - 45***

Now, we have an equation with only one unknown, x. Let's proceed to isolate x. First, subtract 3x from both sides of the equation:

x - 3x = -45***

Combine the x terms:

-2x = -45**

Finally, divide both sides by -2:

x = 22.5*

However, x needs to be an integer, so we made a mistake by assuming x = n. Let's go back to x = 3n - 45.

The question states