"Aaron and Beren are playing a game on an infinite complete binary tree. At the beginning of the game, every edge of the tree is independently labeled A with probability p and B otherwise. Both players are able to inspect all of these labels. Then, starting with Aaron at the root of the tree, the players alternate turns moving a shared token down the tree (each turn the active player selects from the two descendants of the current node and moves the token along the edge to that node). If the token ever traverses an edge labeled B, Beren wins the game. Otherwise, Aaron wins.

What is the infimum of the set of all probabilities p for which Aaron has a nonzero probability of winning the game? Give your answer in exact terms."

From [0]. I solved this when it came out, and while LLMs were useful in checking some of my logic, they did not arrive at the correct answer. Just checked with o3 and still no dice. They are definitely getting closer each model iteration though.

[0] https://www.janestreet.com/puzzles/tree-edge-triage-index/

OpenAI's o4-mini got the right answer after "thinking" for 29 seconds. It's a straightforward puzzle, though: no creativity involved.