The probability that a person takes more than t minutes is 0.12. Therefore, the probability that a person takes t minutes or less is 0.88.

The probability that none of the n people takes more than t minutes is given by:

\((0.88)^n < 0.003\)

Taking logarithms on both sides, we have:

\(n \log(0.88) < \log(0.003)\)

Solving for n gives:

\(n > \frac{\log(0.003)}{\log(0.88)}\)

Calculating the right-hand side:

\(n > 45.4\)

Since n must be an integer, the smallest possible value is:

\(n = 46\)