How many times must you throw two dice to be sure that you get the same sum at least six times?

To ensure that you get the same sum at least six times when throwing two dice, you can use the pigeonhole principle. There are 11 possible sums (from 2 to 12). If you throw the dice 55 times, then by the pigeonhole principle, at least one sum must occur at least six times, because (55 \div 11 = 5) remainder 0, meaning each sum can occur up to five times before hitting the sixth. Thus, you need to throw the dice 55 times to guarantee that the same sum appears at least six times.