Birthday Line — Best Position
You want enough people ahead of you to create many possible birthday matches, but not so many that someone else is likely to have already won.
To win at position k, the first k−1 people must all have different birthdays, and then your birthday must match one of theirs.
At position 20, there are 19 earlier birthdays you could match. More people ahead means more possible matching birthdays.
Every extra person ahead also creates another chance that a duplicate birthday appears before your turn, ending the game before you can win.
Let P(k) be your chance of winning from position k. Comparing adjacent positions:
Moving one spot later helps exactly when k² − k − 365 < 0. The positive root is about 19.61, so the probability rises through position 20 and falls after that.