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500 can cube over 400 - a lot over.
as i calculate it its 346, its the only cube number that's bigger than 200 but smaller than 400
If you start with 401, then the answer to 401^3 is sure to be bigger than 400.
There are two integer answers : 6 and 7. Any value between 5.848 (cube root of 200) and 7.368 (cube root of 400) will satisfy the question.
The number 6 has a cube (6 x 6 x 6) equal to 216. The number 7 has a cube (7 x 7 x 7) equal to 343.