There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?

For an equilateral triangle to be formed from equally spaced points on a circle, the total number of points (N) must be a multiple of 3. If N is a multiple of 3, the vertices of an equilateral triangle must be separated by N/3 points along the circumference. In this case, N = 24. Since 24 is a multiple of 3, equilateral triangles can be formed. The separation between vertices will be 24 / 3 = 8 points. Let's label the points from 0 to 23. An equilateral triangle can be formed by picking a point (say, point 'i'), and then the other two vertices will be at (i + 8) and (i + 16), all modulo 24. Distinct triangles are formed by choosing starting points from 0 up to (N/3 - 1). So, the distinct starting points are 0, 1, 2, 3, 4, 5, 6, 7. 1. (0, 8, 16) 2. (1, 9, 17) 3. (2, 10, 18) 4. (3, 11, 19) 5. (4, 12, 20) 6. (5, 13, 21) 7. (6, 14, 22) 8. (7, 15, 23) If we start with point 8, we get (8, 16, 24 mod 24) = (8, 16, 0), which is the same as the first triangle (0, 8, 16). Therefore, there are 8 distinct equilateral triangles. Analyzing the options: A) 4: Incorrect. B) 6: Incorrect. C) 8: Correct, as calculated. D) 12: Incorrect. The final answer is C.