An equilateral triangular plate is to be cut into n number of identical small equilateral triangular plates. Which one of the following can be possible value of n?

To cut an equilateral triangular plate into 'n' identical small equilateral triangular plates, the number 'n' must be a perfect square. This is because if the side of the large triangle is divided into 'k' equal segments, the total number of small identical equilateral triangles formed will be k^2. Let's examine the given options: A) 196 = 14^2. This is a perfect square. B) 216. This is not a perfect square (14^2 = 196, 15^2 = 225). C) 256 = 16^2. This is a perfect square. D) 296. This is not a perfect square (17^2 = 289, 18^2 = 324). Both 196 and 256 are perfect squares, meaning they are mathematically possible values for 'n' under the general rule. However, in such questions, sometimes a specific method of division is implicitly assumed. A common method is recursive division: dividing an equilateral triangle into 4 smaller identical equilateral triangles by connecting the midpoints of its sides. If this process is repeated, the total number of smallest triangles will be a power of 4 (e.g., 4, 16, 64, 256, etc.). Let's check which of the perfect square options is also a power of 4: - 196 is not a power of 4 (4^3 = 64, 4^4 = 256). - 256 is 4^4 (or 16^2). This is a power of 4. Therefore, considering the possibility of recursive division, 256 is the most appropriate answer among the given options. The final answer is C.