Question

What sample size is needed for the t-test to have a power of 0.8 with an effect size of 0.5?

This is a toy example, since you can just use a pakcage to calculate it, but the simulation approach works for everything, including arbitrarily complex experimental designs, correcting for multiple comparisons and so on(assuming infinite computational resources and you have some prior knowledge of the likely distribution of simulation parameters).

# Run nresps simulations
# The power is simply the fraction of reps where
# the p-value is less than 0.05

nreps = 10000
d = 0.5

n = 50
power = 0
while power < 0.8:
    n1 = n2 = n
    x = np.random.normal(0, 1, (n1, nreps))
    y = np.random.normal(d, 1, (n2, nreps))
    t, p = st.ttest_ind(x, y)
    power = (p < 0.05).sum()/nreps
    print n, power
    n += 1

50 0.7002
51 0.706
52 0.7119
53 0.7181
54 0.7344
55 0.7351
56 0.7405
57 0.7583
58 0.761
59 0.7647
60 0.775
61 0.7878
62 0.7865
63 0.7913
64 0.8004