Final standings with tie-breaks for a club Swiss tournament (Buchholz Cut-1 with virtual opponents, Sonneborn-Berger, direct encounter)

# Final standings with tie-breaks for a club Swiss tournament `players.csv` (name, rating) and `results.csv` (round, white, black, result; rows in any order) hold a finished event. Write `standings.py` (Python 3 stdlib only) that, run as `python standings.py` in a directory with those two files, writes `standings.csv`. We also run it on other events, so do not hard-code anything. ## Results 1. `result` is `1-0`, `0-1`, `1/2-1/2` (games played over the board), `+-` / `-+` (forfeit: white / black gets 1 point, the other 0; **not** played over the board), or, with `black` empty, `BYE` (white gets 1 point) or `HALF` (white gets 1/2). A player with no row in a round got 0 points that round. R = the highest round number in the file. 2. **Points** = sum of points from all rows. ## Tie-breaks (higher is better at every step) 3. **Buchholz Cut-1 (`bh_cut1`)**: for each round 1..R take a contribution: the opponent's final points if the player played that round **over the board**, otherwise (bye, half bye, forfeit win or loss, absent) the player's **own** final points. Sum all R contributions and subtract the single smallest one. 4. **Sonneborn-Berger (`sb`)**: sum over games played over the board of opponent's final points x points the player scored in that game (1, 1/2 or 0). Unplayed rounds add nothing. 5. Rank by points, then bh_cut1, then sb. 6. **Direct encounter**: players still equal on all of points, bh_cut1 and sb form a group. If the group has 2+ players **and every pair in it played each other over the board** (in some round), order them by points scored in over-the-board games against other group members. Otherwise this step does nothing for the group. 7. Remaining ties: more over-the-board wins (forfeit wins do not count), then higher rating, then name (ascending, plain string order). 8. Use exact arithmetic (all values are multiples of 1/4). ## Output 9. `standings.csv`: header `rank,name,points,bh_cut1,sb,wins`; ranks 1..n without gaps (every player gets a distinct rank); `points` and `bh_cut1` with 1 decimal, `sb` with 2 decimals, `wins` = over-the-board wins as an integer. Every player in `players.csv` is listed. ## Scoring Hidden checks (partial credit = fraction passed; all must pass), each comparing `standings.csv` exactly: `fixture_exact` (this directory's files), `forfeits_and_byes` (1-2, 7), `buchholz_virtual` (3), `sonneborn_berger` (4), `direct_encounter` (6), `final_tiebreaks` (7), `fresh_event_exact` (a freshly generated event). Each run has a 30-second limit.