Binomial Probability of At Least One Success

P(at least one 6 in n throws) = 1 − P(no 6 in n throws) = 1 − (5/6)ⁿ.
 1 − (5/6)ⁿ > 0.9 ⟹ (5/6)ⁿ < 0.1.
Taking natural log: n × ln(5/6) < ln(0.1) ⟹ n > ln(0.1)/ln(5/6) (inequality flips since ln(5/6) < 0).
 ln(0.1) ≈ −2.3026, ln(5/6) ≈ −0.1823. So n > 2.3026/0.1823 ≈ 12.63.
 The smallest integer n satisfying this is n = 13.