What is the smallest positive perfect cube that can be written as the sum of three consecutive integers?
A perfect cube is a number that can be written as the cube of some integer, like 8 = 2^3 or 27 = 3^3. I want to find the smallest one that can also be written as the sum of three consecutive integers, like 3 + 4 + 5 = 12. Let me try to write this equation algebraically. If x is the first of the three consecutive integers, then the next two are x + 1 and x + 2. So the sum is x + (x + 1) + (x + 2) = 3x + 3. I want this sum to be equal to some perfect cube, say y^3. So I have 3x + 3 = y^3. Now I need to find some values of x and y that make this equation true, and I want the smallest possible y, since that will give me the smallest perfect cube. I can try some small values of y and see what x they give me. If y = 1, then x + 1 = 1/3, which is not an integer, so that doesn't work. If y = 2, then x + 1 = 8/3, which is also not an integer, so that doesn't work either. If y = 3, then x + 1 = 9, which is an integer, so that works! That means x = 8, and the three consecutive integers are 8, 9, and 10. Their sum is 27, which is also 3^3, so this is the smallest positive perfect cube that can be written as the sum of three consecutive integers. 