Question

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

Answer 1

17 times

Answer 2

The sum can be an integer between 3 (1+1+1) and 18(6+6+6). So, it can take 16 distinct values. So the dice should be rolled 16+1 = 17 times to ensure the same sum is rolled twice(as is correctly stated in the reply above :) )

This follows from the Pigeonhole Principle which is a fancy same for the common sense occurrence that if n items are distributed among m containers and n > m , then at least one of the containers will contain more than one item.

In this particular problem, the number of possible distinct sums (16) is the number of containers and we are asked to find out the minimum number of items to ensure that at least one of the containers contains more than one item. If we roll the dice 16 times , it can happen that each time we get a different sum but if we roll it once more, then this time the sum we get has to be similar to one of the 16 sums we got earlier.

Now, say if the question asked how many times we should roll the dice to ensure we get the same sum 7 times, we would proceed similarly.
In that case if we roll the dice (6*16 +1) = 97 times we would ensure the same sum 7 times.

More generally, if n = km + 1 objects are distributed among m containers , one of the containers will contain at least k+1 items.