What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice?

For three fair six-sided dice, the possible sum of the faces rolled can be any digit from 3 to 18.

For instance the minimum sum occurs when all three dices shows 1 (i. e. 1 + 1 + 1 = 3) and the maximum sum occurs when all three dces shows 6 (i. e. 6 + 6 + 6 = 18).

Thus, there are 16 possible sums when three six-sided dice are rolled.

Therefore, from the pigeonhole principle, the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice is 16 + 1 = 17 times.

The pigeonhole principle states that if n items are put into m containers, with n > m > 0, then at least one container must contain more than one item.

That is for our case, given that there are 16 possible sums when three six-sided dice is rolled, for there to be two same sums, the number of sums will be greater than 16 and the minimum number greater than 16 is 17.