Cube (algebra) however, easy to show that most numbers are not perfect cubes because "all" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: The equation has
Doubling the cube Doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible using only a compass and straightedge, but even in ancient times solutions were known that employed other tools. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made many futile attempts