I added a new problem (#29 of now 83 problems) to my dice problem collection.
If we roll a die and multiply the results as we go, how many rolls on average will we need until the product is a perfect square?
For a six-sided die the expected number of rolls is 8. Curiously, this is the same as for a five-sided die. The general situation is that the expected number of 2^pi(s) where s is the number of sides of the die and pi(x) is the number of primes less than or equal to x.
Since 6 is not prime, the expected number is the same for 5 and 6-sided dice.
Curiously, it does not matter if the die is fair. As long as, for example, all sides are possible, the expectation is 2^pi(s).
One could ask: for what subsets A of the positive integers does the expected number of rolls until the product is in A depend only on s? This is the case when A is the set of perfects squares. What other A can we consider?
