<p>Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is __________.</p>
Step-by-Step Solution
Key Concept: If a triangle has n balls per side, it contains n(n+1)/2 total balls. Setting up the equation: n(n+1)/2 + 99 = (n-2)², where n-2 is the side length of the square, yields the constraint that links triangular and square numbers.
<p><strong>Step 1:</strong> Let the triangle have n rows. Then total balls in triangle = 1 + 2 + 3 + ... + n = n(n+1)/2</p><p><strong>Step 2:</strong> The square is formed by adding 99 balls, so square has n(n+1)/2 + 99 balls. If the square has side length m balls per side, then total = m²</p><p><strong>Step 3:</strong> Given condition: m = n - 2 (each side of square has 2 balls less than triangle's side). Therefore: n(n+1)/2 + 99 = (n-2)²</p><p><strong>Step 4:</strong> Expanding: n(n+1)/2 + 99 = n² - 4n + 4</p><p>n²/2 + n/2 + 99 = n² - 4n + 4</p><p>n/2 + 4n + 99 = n²/2 + 4</p><p>9n/2 + 95 = n²/2</p><p>9n + 190 = n²</p><p>n² - 9n - 190 = 0</p><p><strong>Step 5:</strong> Using quadratic formula: n = (9 ± √(81 + 760))/2 = (9 ± √841)/2 = (9 ± 29)/2</p><p>n = 19 or n = -10 (rejected)</p><p><strong>Step 6:</strong> Total balls in triangle = n(n+1)/2 = 19(20)/2 = 190</p><p><strong>Verification:</strong> Square has 190 + 99 = 289 = 17² balls, and 17 = 19 - 2 ✓</p><p>∴ Answer: <strong>190</strong></p>
Correct Answer: 190