Roll me a cube!

There are four three digit cube numbers containing only the digits $1\text{ to } 6$ :

• $5^3 = 125$
• $6^3 = 216$
• $7^3 = 343$
• $8^3 = 512$

So, you must roll one of the following combinations of digits:

• $1,2,5$ - Six ways
• $2,1,6$ - Six ways
• $3,4,3$ - Three ways

Total: $15$ Ways

And there are $6^3 = 216$ ways to roll the three dice.

So, the probability is:

$P = \dfrac{15}{216} = \dfrac{5}{72}$

$5+72 = \boxed{77}$