The sum of the squares of three consecutive integers is 365. What is the middle integer?
["Title: Find the Middle Integer in Three Consecutive Numbers Whose Squares Sum to 365", "When faced with a number puzzle like “The sum of the squares of three consecutive integers is 365. What is the middle integer?”, solving it combines logic, algebra, and arithmetic. Let’s break down the problem step by step to uncover the solution and understand how it fits into broader mathematical thinking.", "### Understanding the Problem", "We are looking for three consecutive integers — let’s call them:\n- ( x - 1 )\n- ( x ) (the middle integer)\n- ( x + 1 )", "The condition is that the sum of their squares equals 365:", "[\n(x - 1)^2 + x^2 + (x + 1)^2 = 365\n]", "### Expanding the Expressions", "Expand each squared term:", "- ( (x - 1)^2 = x^2 - 2x + 1 )\n- ( x^2 = x^2 )\n- ( (x + 1)^2 = x^2 + 2x + 1 )", "Now, sum them up:", "[\n(x^2 - 2x + 1) + x^2 + (x^2 + 2x + 1)\n]", "Combine like terms:", "- ( x^2 + x^2 + x^2 = 3x^2 )\n- ( -2x + 2x = 0 ) (the linear terms cancel)\n- ( 1 + 1 = 2 )", "So the equation simplifies to:", "[\n3x^2 + 2 = 365\n]", "### Solving the Equation", "Subtract 2 from both sides:", "[\n3x^2 = 363\n]", "Divide both sides by 3:", "[\nx^2 = 121\n]", "Take the square root:", "[\nx = \pm 11\n]", "Since we are dealing with integers and the sum of squares is positive, both ( x = 11 ) and ( x = -11 ) are mathematically valid. But since the question asks for the middle integer and typically implies a positive context, we focus on the positive case — especially when talking about “the” solution in such puzzles.", "### Verifying the Solution", "Check with ( x = 11 ):", "- Integers: ( 10, 11, 12 )\n- Squares: ( 100 + 121 + 144 = 365 ) ✅\n- Middle integer: ( 11 )", "Check with ( x = -11 ):", "- Integers: ( -12, -11, -10 )\n- Squares: ( 144 + 121 + 100 = 365 ) ✅\n- Middle integer: ( -11 )", "Both sets satisfy the equation — but when solving for a middle integer in a real-world context without specific sign constraints, both are acceptable. However, because 11 is the standard positive answer expected in such puzzles, and the middle number is clearly identified as 11, we conclude:", "### Final Answer", "The middle integer is 11.", "### Why This Problem Matters (SEO Implications)", "This type of problem appeals to students and math enthusiasts who enjoy pattern recognition, algebraic manipulation, and logical reasoning. The widespread interest in number theory puzzles, combined with clear step-by-step solutions like this one, makes the article highly SEO-friendly. Including keywords such as:\n- “sum of squares of three consecutive integers”\n- “middle integer puzzle”\n- “consecutive integers problem with square sum”\n– boosts visibility for researchers, students, and casual learners alike.", "Leveraging such engaging content attracts engaged audiences and improves search rankings in educational and math-related niches.", "---", "Bottom Line:\nGiven the condition that the sum of the squares of three consecutive integers equals 365, the middle integer is 11."]
