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. 
What is the smallest positive perfect cube that can be written as the sum of three consecutive integers?