{"task": {"agent_timeout": 600, "task": "omnimath_105", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nLet $n$ be a positive integer. Initially, a $2n \\times 2n$ grid has $k$ black cells and the rest white cells. The following two operations are allowed :\n(1) If a $2\\times 2$ square has exactly three black cells, the fourth is changed to a black cell; \n(2) If there are exactly two black cells in a $2 \\times 2$ square, the black cells are changed to white and white to black.\n Find the smallest positive integer $k$ such that for any configuration of the $2n \\times 2n$ grid with $k$ black cells, all cells can be black after a finite number of operations.\n\n## Instructions\n\nSolve the mathematical problem above and write your final answer to `/workspace/answer.txt`.\n\n**Important**: Write only your final answer to the file, not the full solution process.\n\n### Guidelines\n\n- Provide your final numerical answer or mathematical expression\n- Write the answer as plain text (no special formatting needed)\n- Be precise and clear in your answer\n- The answer should directly respond to what the problem asks for\n\n### Example\n\nIf the problem asks \"What is 2 + 2?\", your answer file should contain:\n\n```\n4\n```\n\nYour answer will be evaluated against the correct solution using an automated grading system.\n", "memory": "", "runnable": false, "difficulty": "hard", "language": "", "cpus": "", "instruction_truncated": false, "category": "math", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "omnimath", "tags": ["omni-math"]}, "runs": []}