{ "slug": "n-queens", "title": "Backtracking — N-Queens", "category": "Recursion & DP", "difficulty": "hard", "complexity": { "time": "O(n!)", "space": "O(n)" }, "idea": "Place one queen per row. For each square check whether an earlier queen already attacks it down a column or a diagonal. If a row runs out of safe squares, the previous choice was wrong — take that queen back and try the next square. That retreat is backtracking.", "code": [ "solve(row):", " if row == n: solved", " for col in 0..n-1:", " if safe(row, col): place; if solve(row+1): return true; remove", " return false # backtrack" ], "example": { "array": [] }, "invariant": "Each step is the real data structure at that step — the picture is a pure function of it.", "stepCount": 18, "steps": [ { "i": 0, "op": "grid", "caption": "Place 5 queens so none share a column or diagonal — one queen per row.", "note": null, "codeLine": 1, "stats": null, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 1, "op": "grid", "caption": "Row 0, column 0 is safe — place a queen.", "note": null, "codeLine": 4, "stats": { "placed": 1 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "insert", null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 2, "op": "grid", "caption": "Row 1, column 0 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 1 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ "remove", null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 3, "op": "grid", "caption": "Row 1, column 1 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 1 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ null, "remove", null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 4, "op": "grid", "caption": "Row 1, column 2 is safe — place a queen.", "note": null, "codeLine": 4, "stats": { "placed": 2 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, "insert", null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 5, "op": "grid", "caption": "Row 2, column 0 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 2 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ null, null, null, null, null ], [ "remove", null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 6, "op": "grid", "caption": "Row 2, column 1 is attacked by the queen in row 1.", "note": null, "codeLine": 4, "stats": { "placed": 2 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, "compare", null, null ], [ null, "remove", null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 7, "op": "grid", "caption": "Row 2, column 2 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 2 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ null, null, null, null, null ], [ null, null, "remove", null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 8, "op": "grid", "caption": "Row 2, column 3 is attacked by the queen in row 1.", "note": null, "codeLine": 4, "stats": { "placed": 2 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, "compare", null, null ], [ null, null, null, "remove", null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 9, "op": "grid", "caption": "Row 2, column 4 is safe — place a queen.", "note": null, "codeLine": 4, "stats": { "placed": 3 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, "insert" ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 10, "op": "grid", "caption": "Row 3, column 0 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 3 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, null, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ "remove", null, null, null, null ], [ null, null, null, null, null ] ] }, { "i": 11, "op": "grid", "caption": "Row 3, column 1 is safe — place a queen.", "note": null, "codeLine": 4, "stats": { "placed": 4 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, "insert", null, null, null ], [ null, null, null, null, null ] ] }, { "i": 12, "op": "grid", "caption": "Row 4, column 0 is attacked by the queen in row 0.", "note": null, "codeLine": 4, "stats": { "placed": 4 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ "compare", null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ "remove", null, null, null, null ] ] }, { "i": 13, "op": "grid", "caption": "Row 4, column 1 is attacked by the queen in row 3.", "note": null, "codeLine": 4, "stats": { "placed": 4 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, "compare", null, null, null ], [ null, "remove", null, null, null ] ] }, { "i": 14, "op": "grid", "caption": "Row 4, column 2 is attacked by the queen in row 1.", "note": null, "codeLine": 4, "stats": { "placed": 4 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, null, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, "compare", null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, "remove", null, null ] ] }, { "i": 15, "op": "grid", "caption": "Row 4, column 3 is safe — place a queen.", "note": null, "codeLine": 4, "stats": { "placed": 5 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, 1, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "♛", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, "insert", null ] ] }, { "i": 16, "op": "grid", "caption": "Every row filled — this is a valid arrangement.", "note": null, "codeLine": 2, "stats": { "placed": 5 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, 1, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "♛", "" ] ], "focus": [ [ "settled", null, null, null, null ], [ null, null, "settled", null, null ], [ null, null, null, null, "settled" ], [ null, "settled", null, null, null ], [ null, null, null, "settled", null ] ] }, { "i": 17, "op": "grid", "caption": "Solved with 0 backtracks. Columns by row: 0, 2, 4, 1, 3.", "note": null, "codeLine": 2, "stats": { "placed": 5 }, "rows": 5, "cols": 5, "rowLabels": [ "0", "1", "2", "3", "4" ], "colLabels": [ "0", "1", "2", "3", "4" ], "grid": [ [ 1, null, null, null, null ], [ null, null, 1, null, null ], [ null, null, null, null, 1 ], [ null, 1, null, null, null ], [ null, null, null, 1, null ] ], "labels": [ [ "♛", "", "", "", "" ], [ "", "", "♛", "", "" ], [ "", "", "", "", "♛" ], [ "", "♛", "", "", "" ], [ "", "", "", "♛", "" ] ], "focus": [ [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ], [ null, null, null, null, null ] ] } ] }