Inspired by this question.
A wealthy, famous eccentric had a huge cube of equal sized diamonds.* Due to her eccentricities the number of diamonds was known to be both palindromic in base ten, and a multiple of $7$ - her lucky number.
* That is a cube of $n^3 | n \in \Bbb{N}$ diamonds.
One day $7$ thieves stole the cube and got away in a spaceship. They planned to break the cube up to share the diamonds out equally, since it would be too difficult to sell such a unique piece anywhere in the system.
During the flight $2$ of the thieves decided to do away with the other $5$ thieves and share the diamonds equally, however they found that $1$ diamond would be left over.
As they discovered this a third thief discovered them, and suggested they do away with the other $4$ thieves instead and again share the diamonds equally. Again there would be $1$ diamond left over, and again as they discovered this a fourth thief discovered them.
The process repeated until the seventh thief found the others plotting against him, and still with a single remaining diamond after splitting them into $6$ equal shares.
What was the minimum number of diamonds in the cube?