# Roll a cube with a tetradron?

**Discrete Mathematics**Level 5

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.

**Image credit:** www.midlamminiatures.co.uk

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