<p>How many different squares are there on a chessboard?</p>
Step-by-Step Solution
Key Concept: Squares of different sizes can be formed on a chessboard—not just 1×1 squares. You must count all k×k squares for k = 1, 2, 3, ..., 8 and sum them using the formula for each size.
<p><strong>Step 1:</strong> On an 8×8 chessboard, a k×k square can be placed in (9−k)² different positions (since the top-left corner can be placed in (9−k) rows and (9−k) columns).</p><p><strong>Step 2:</strong> Count squares of each size:</p><ul><li>1×1 squares: (9−1)² = 8² = 64</li><li>2×2 squares: (9−2)² = 7² = 49</li><li>3×3 squares: (9−3)² = 6² = 36</li><li>4×4 squares: (9−4)² = 5² = 25</li><li>5×5 squares: (9−5)² = 4² = 16</li><li>6×6 squares: (9−6)² = 3² = 9</li><li>7×7 squares: (9−7)² = 2² = 4</li><li>8×8 squares: (9−8)² = 1² = 1</li></ul><p><strong>Step 3:</strong> Total = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 8² + 7² + 6² + 5² + 4² + 3² + 2² + 1²</p><p><strong>Step 4:</strong> Using the formula Σk² = n(n+1)(2n+1)/6 for n=8: [8(9)(17)]/6 = 1224/6 = 204</p><p>∴ Answer: <strong>204</strong></p>
Correct Answer: 204