A fair die is to be rolled several times. What is the SMALLEST number of rolls for which the probability of getting at least one six is more than 1/2?

Work with the complement: no six in $n$ rolls has probability (5/6)ⁿ, so at least one six has probability 1 − (5/6)ⁿ, and we need (5/6)ⁿ to fall below 1/2. Successively, $(5/6{)}^2=25/36$ ≈ 0.694, $(5/6{)}^3=125/216$ ≈ 0.579 and $(5/6{)}^4=625/1296$ ≈ 0.482. Only at $n=4$ does it drop below a half, giving 1 − 625/1296 = 671/1296 ≈ 0.518. So 4 rolls are needed. Three rolls fall just short at about 0.421, which is why guessing 3 is so tempting. Two rolls give only 11/36 ≈ 0.306. The value 6 comes from the belief that since a six appears on average once in six rolls, six rolls make it an even chance — but the chance of at least one six in six rolls is about 0.665, well past a half, and the average tells you nothing directly about the probability.