Puzzle: Cubic Sums |

An elliptic curve is generally an order 3 (cubic) equation. The animation below shows the Thompson
cubic, but that's too hard for a general puzzle. Instead, many good problems are in cubic numbers: 1^3 = 1,
2^3 = 8, 3^3 = 27, leading to the sequence 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744... The negative cubes start -1, -8, -27, ...

Easy: Find a non-zero number expressible as a sum of two positive cubes in two different ways.

Moderate: Find a non-zero number expressible as a sum of two cubes in three different ways.

Too hard: Find four different non-zero numbers so that the sum of any pair is a cubic number.

Easy: Find a non-zero number expressible as a sum of two positive cubes in two different ways.

Moderate: Find a non-zero number expressible as a sum of two cubes in three different ways.

Too hard: Find four different non-zero numbers so that the sum of any pair is a cubic number.