Sums of three cubes: 114, and a fourth way to write 3

Write 114, the smallest open case, as a sum of three integer cubes, or find a fourth representation of 3 as a sum of three cubes.

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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "number-theory"
	5	+ ],
	6	+ "type": "construction",
	7	+ "source": {
	8	+ "key": "frontiermath-open",
	9	+ "name": "FrontierMath: Open Problems",
	10	+ "url": "https://epoch.ai/frontiermath/open-problems/sum-of-three-cubes",
	11	+ "id": "sum-of-three-cubes",
	12	+ "curator": "Epoch AI",
	13	+ "status": "unsolved",
	14	+ "as_of": "2026-10-20",
	15	+ "catalog": "0521c19bbd38"
	16	+ }
	17	+}