{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261027-r07-graph-walk-simple-path-0083", "pilot_number": "R0025-13358", "number": 13358, "family_id": "qf-mod27-walk-repetition-classification", "title": { "zh-Hant": "第 13358 題：圖walk與simple path", "en": "Question 13358: Distinguish Repeated Vertices, Repeated Edges and Closed Walks" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261027-r07-graph-walk-simple-path-0083", "batch_id": "batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-graph-walk-simple-path" ], "title_zh": "圖walk與simple path", "stem_zh": "有限無向無loop簡單圖與完整頂點序列。操作=walk-repetition-classification；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"vertices\":[3,2,0,1,0]}。請依 [walk,edge_length,repeated_vertices,repeated_edges,trail,simple_path,closed] 欄序回答。", "answer": { "kind": "expression", "value": "(是, 4, [0], [[0, 1]], 否, 否, 否)", "explanation_zh": "Wave4根節點核對=(是, 4, [0], [[0, 1]], 否, 否, 否)；依完整明示資料重算；完整有限／精確座標切片不自動升格為任意無限定理；每對連續頂點都須相鄰且長度是邊數；walk可重複但simple path不得重複頂點。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "圖walk與simple path" ] }, "variant": { "family_id": "qf-mod27-walk-repetition-classification", "seed": 202610277083, "parameters": { "n": 6, "edges": [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 0, 4 ], [ 0, 5 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 3, 4 ], [ 3, 5 ], [ 4, 5 ] ], "vertices": [ 3, 2, 0, 1, 0 ], "domain": "Complete_finite_undirected_walk", "operation": "walk-repetition-classification", "semantic_template": "qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、一般完備性、內積／有界性、mutual independence、directed graph、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261027-r07-graph-walk-simple-path-0083", "batch_id": "batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "04b863a17621f54f55843d61c0b30638017968f9cc0db6b59f1b0d344f00f4a7", "family_id": "qf-mod27-walk-repetition-classification", "seed": 202610277083, "semantic_template": "qf-mod27-walk-repetition-classification/{\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"n\":6,\"vertices\":[3,2,0,1,0]}" }, "fields": { "title": "Distinguish Repeated Vertices, Repeated Edges and Closed Walks", "stem": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.", "answer": { "kind": "expression", "value": "(Yes, 4, [0], [[0, 1]], No, No, No)", "explanation": "Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No)." }, "skills": [ "Distinguish Repeated Vertices, Repeated Edges and Closed Walks" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "有限無向無loop簡單圖與完整頂點序列。操作=walk-repetition-classification；資料={\"n\":6,\"edges\":[[0,1],[0,2],[0,3],[0,4],[0,5],[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],\"vertices\":[3,2,0,1,0]}。請依 [walk,edge_length,repeated_vertices,repeated_edges,trail,simple_path,closed] 欄序回答。", "en": "Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[3, 2, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "carrier Fin(6);raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];original vertex sequence=[3, 2, 0, 1, 0]", "after": "canonical undirected simple edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];nonempty typed sequence retained", "relation": "implies", "rule_id": "walk_simple_graph_and_ordered_candidate", "conditions": [], "explanation": { "zh-Hant": "圖邊正規化無向端點並拒絕loop與反向重複；頂點序列則保留順序與重複，重複不是輸入解析錯，而可能使walk或simple資格失敗。", "en": "Normalize undirected graph edges and reject loops/reversed duplicates.Preserve vertex-sequence order and repetitions.Repetition is not a parsing error in a candidate,though it may fail walk or simplicity qualification." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "proposed step index=0;from=3;to=2;unordered step edge=[2, 3]", "after": "edge present=是", "relation": "implies", "rule_id": "walk_every_consecutive_pair_not_just_vertices", "conditions": [], "explanation": { "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。", "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "proposed step index=1;from=2;to=0;unordered step edge=[0, 2]", "after": "edge present=是", "relation": "implies", "rule_id": "walk_every_consecutive_pair_not_just_vertices", "conditions": [], "explanation": { "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。", "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "proposed step index=2;from=0;to=1;unordered step edge=[0, 1]", "after": "edge present=是", "relation": "implies", "rule_id": "walk_every_consecutive_pair_not_just_vertices", "conditions": [], "explanation": { "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。", "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "proposed step index=3;from=1;to=0;unordered step edge=[0, 1]", "after": "edge present=是", "relation": "implies", "rule_id": "walk_every_consecutive_pair_not_just_vertices", "conditions": [], "explanation": { "zh-Hant": "每兩個連續頂點都須在原圖相鄰；立即重複頂點給對角步而圖無loop，所以該步失敗。方向不改無向邊身份。", "en": "Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "all bad proposed steps=[]", "after": "walk=是", "relation": "implies", "rule_id": "complete_walk_candidate_qualification", "conditions": [], "explanation": { "zh-Hant": "完整步清單全通過才walk；單頂點無步清單空泛通過，仍是長度0的合法walk。", "en": "The whole step list must pass.A singleton has no proposed steps and passes vacuously as a valid length-zero walk." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "proposed edge length := sub(5, 1)", "after": "proposed edge length = 4", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色及已核中間量精確運算；保留係數、座標、次序與正負號。", "en": "Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 proposed edge length，不以小數近似替代。", "en": "Recompute proposed edge length with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "5", "1" ], "output": "4", "label": "proposed edge length" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "original vertex sequence=[3, 2, 0, 1, 0]", "after": "sorted distinct repeated vertex labels=[0]", "relation": "implies", "rule_id": "simple_path_vertex_repetition_full_sequence", "conditions": [], "explanation": { "zh-Hant": "simple path禁止任何重複頂點，不只相鄰重複；閉walk的起終點重複亦在內。", "en": "A simple path forbids any repeated vertex,not only immediate repeats.This includes a closed walk's repeated endpoint." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "proposed step index=0;unordered edge=[2, 3]", "after": "seen at earlier step=否;distinct repeated edges so far=[]", "relation": "implies", "rule_id": "trail_undirected_edge_first_repeat_order", "conditions": [], "explanation": { "zh-Hant": "把每個候選步都視為無向邊後判重，反向折返仍重複同一邊；失敗清單按首次重複遇到的順序，不按邊標籤排序。", "en": "Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "proposed step index=1;unordered edge=[0, 2]", "after": "seen at earlier step=否;distinct repeated edges so far=[]", "relation": "implies", "rule_id": "trail_undirected_edge_first_repeat_order", "conditions": [], "explanation": { "zh-Hant": "把每個候選步都視為無向邊後判重，反向折返仍重複同一邊；失敗清單按首次重複遇到的順序，不按邊標籤排序。", "en": "Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "proposed step index=2;unordered edge=[0, 1]", "after": "seen at earlier step=否;distinct repeated edges so far=[]", "relation": "implies", "rule_id": "trail_undirected_edge_first_repeat_order", "conditions": [], "explanation": { "zh-Hant": "把每個候選步都視為無向邊後判重，反向折返仍重複同一邊；失敗清單按首次重複遇到的順序，不按邊標籤排序。", "en": "Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "proposed step index=3;unordered edge=[0, 1]", "after": "seen at earlier step=是;distinct repeated edges so far=[[0, 1]]", "relation": "implies", "rule_id": "trail_undirected_edge_first_repeat_order", "conditions": [], "explanation": { "zh-Hant": "把每個候選步都視為無向邊後判重，反向折返仍重複同一邊；失敗清單按首次重複遇到的順序，不按邊標籤排序。", "en": "Detect repetitions after normalizing each proposed step as an undirected edge.Backtracking repeats the same edge.Report distinct repeats in first-repeat encounter order,not sorted edge order." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "walk=是;length=4;start=3;end=0;repeated vertices=[0];repeated edges=[[0, 1]]", "after": "trail=否;simple path=否;closed positive-length walk=否", "relation": "implies", "rule_id": "walk_trail_simple_closed_separate_obligations", "conditions": [], "explanation": { "zh-Hant": "trail不重邊，simple path不重點，closed另需正長度且首尾相同；三者不是同義詞。單頂點是trail與simple path，但此協定下不算closed；closed也不是simple cycle證書。", "en": "A trail has no repeated edges,a simple path no repeated vertices,and closed additionally requires positive length and equal endpoints.These are distinct.A singleton is a trail/simple path but is not closed in this protocol;closed is not a simple-cycle certificate." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ true, 4, [ 0 ], [ [ 0, 1 ] ], false, false, false ] }, "conclusion": { "canonical_value": "(是, 4, [0], [[0, 1]], 否, 否, 否)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations/questions.jsonl", "batch_sha256": "67197261606ced0d60fcec689034546dca308648cd0928f12df5674ba363e5bb", "original_line": 83, "original_line_sha256": "5df0f6f61e50d31f7587423feb39cfb6861010541a6425854a42b74062e98e79", "question_compact_sha256": "04b863a17621f54f55843d61c0b30638017968f9cc0db6b59f1b0d344f00f4a7", "english_row_compact_sha256": "0275e6bdf81c2ed0f7525431f44da5490b1df3608d4ae7254800df49bf9bde71", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations.en.jsonl" }, "mko_bindings": { "mko-graph-walk-simple-path": { "sha256": "e48ce83c081f9aa69ead3069dd9ed3c1f58b44d320e7dc35d7794f060c83b554", "english_sha256": "26c10dfdc30c28816aa973a1410f93d152f298fc836343c29bbeab96045bacd5", "original_path": "public/data/mko/graph-walk-simple-path.json", "english_path": "public/data/i18n/editions/mko-graph-walk-simple-path.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a02aa2dccf790869cf864b821d0e60119686214561153daba232d921e07e7924", "original_candidate_sha256": "770aa55f38603fa31d619296b87625a3b558d32675aa6e704191e27cb7a012a6", "original_candidate_line_sha256": "770aa55f38603fa31d619296b87625a3b558d32675aa6e704191e27cb7a012a6" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-007-p26", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5", "question_first_witness_sha256": "ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 13, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9c5720965f9f7eef7eba62c265ac11746ce7feb77435a68ef3b19b206d35d4fe", "arithmetic_steps": 1 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }