# Roll a cube with a tetradron?

Cubone would like to roll a four sided die with vertices labeled 1 to 4, as shown above until he rolls a string of digits that represent an integer cubed that is less than a million.

So, now for the question: If the expected number of rolls he needs to make to get a string of digits representing a perfect cube, as described above, is $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is $$a+b$$?

Clarification: If, for example, it were an $$8$$-sided die, and he rolled a $$2$$ followed by a $$7$$ he would be done, since $$27$$ is a perfect cube, namely $$3^3$$. Or, if he rolls a $$1$$ any time he is done, since $$1$$ is also a perfect cube.

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