task_games__hex__connection_gap_count

Contract

1. Domain: games
2. Scene: hex
3. Scene id: hex
4. Public task id: task_games__hex__connection_gap_count
5. Supported query_id values: single
6. Answer schema: integer_count
7. Annotation schema: point_set
8. Program schema: count(unique_minimum_empty_cells_to_connect_sides(player, board_state)); scene=hex; scope=connection_gap_count

Program Contract

Program: count(unique_minimum_empty_cells_to_connect_sides(player, board_state)); scene=hex; scope=connection_gap_count

Candidate set: the visible game board, pieces, tokens, cards, tiles, marked state, legal-move cues, result panels, and labeled options inside the connection_gap_count objective scope. Operands: visible scene state and prompt-bound operands named by unique_minimum_empty_cells_to_connect_sides, player, board_state, hex, connection_gap_count. Operation: evaluate count over the candidate set using the visible game state, rules, legal moves, comparisons, counts, simulations, or option-selection constraints encoded in the program expression; generation enforces a unique final answer. Output binding: answer uses the integer schema; generation binds a unique final answer. Annotation witnesses: annotation uses the point_set schema; the prompt/annotation contract defines the minimal visual witnesses. Query ids: single.

Reasoning Operations

Families: counting, ranking, topology

Generation Notes

1. query_id=single is the public no-branch query id; the prompt uses the Hex connection-gap template.
2. The generator rejects boards with multiple distinct minimum gap sets so annotation has a unique witness set.
3. Annotation is the point set for the empty cells in that unique minimum connection gap.