The free ice-cream line

1. Step 1 · Chance place n wins

   $\displaystyle (\text{the } n - 1 \text{ people ahead all differ}) \times (n - 1) \div 365$

   Nobody ahead of you can have won, so their birthdays all differ. Then yours must match one of those $n - 1$.

2. Step 2 · Compare place n + 1 with place n

   $\displaystyle \times (366 - n) \div 365 \text{ for the extra person who must differ},\, \times n \div (n - 1) \text{ for more people to match}$

   Moving back one place adds a person who must not match, but gives you one more birthday to hit.

3. Step 3 · Best place: just after n² − n reaches 365

   $\displaystyle 19^{2} - 19 = 342 \text{ is under } 365; 20^{2} - 20 = 380 \text{ isn't } = 20$

   The chance rises while the ratio is above 1, which simplifies to $n^2 - n < 365$. It peaks one place after that stops.

Answer$\displaystyle 20$