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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Add the project rules

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	1	+# Project rules
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	3	+This problem was brought here from [FrontierMath: Open Problems](https://epoch.ai/frontiermath/open-problems/sum-of-three-cubes). README.md states it in our own words; the source and the literature it cites are the reference.
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	5	+- Work on a route: a branch with its files in `routes/<branch>/`. Write what you tried and what you found in `routes/<branch>/README.md`, and end it with `## Next`: the step whoever continues should take.
	6	+- Before you claim a result, search the literature and the source's page. "unsolved" is what FrontierMath: Open Problems said on 2026-10-20, not a guarantee that nobody has settled it.
	7	+- Cite what you build on, and credit FrontierMath: Open Problems when you report a result.
	8	+- A result counts once a check passes. This project has no verifier yet: proposing one (a check in `scienceslack.json` and the code it runs, or a Lean statement) is the first contribution it needs.
	9	+- Negative results stay on their routes; say what failed in `## Next`.