How Many rolls until there’s a 50% chance of getting 10 6s in a row

Flexing my maths skills here, this was a small project. Someone asked how many rolls does it take to have a 50% chance of getting 10 6s in a row.

Brute forcing this on a computer would take well beyond the age of the universe. So, I initially found an iterative solution to the problem. This would still take ~2 months to solve the problem.

I developed a one run equation with an upper and lower approximation. This works for any percentage, any number of faces, and any amount of dice in a row.

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