{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20260914-basic-counting-principle-0006", "pilot_number": "R0070-35526", "number": 35526, "family_id": "qf-basic-counting-principle", "title": { "zh-Hant": "第 35526 題：基本計數原理邊界", "en": "Question 35526: Boundary of the Basic Counting Principles" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20260914-basic-counting-principle-0006", "batch_id": "batch-20260914-domain-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-counting-principle" ], "title_zh": "基本計數原理邊界", "stem_zh": "一個流程有兩個連續階段；第一階段有 3 種選擇，且每一種之後第二階段固定有 3 種選擇。總組合數是多少？", "answer": { "kind": "numeric", "value": 9, "explanation_zh": "每一個第一階段選擇後都有固定 3 種後續選擇，所以 3×3=9。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "基本計數原理", "條件辨識" ] }, "variant": { "family_id": "qf-basic-counting-principle", "seed": 20260999, "parameters": { "mode": "multiplication_fixed", "branch_counts": [ 7, 8 ], "stages": [ 3, 3 ], "overlap_witness": null, "conditional_stage_witness": null, "semantic_template": "qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,3]}" } }, "verification": { "status": "candidate", "checks": [ "schema_pending", "independent_recomputation_pending" ], "warning_zh": "本題僅依明示的有限參數重算或分類；不構成普遍證明、形式 Evidence、人工審查或發布。" }, "provenance": { "generator_role": "question_producer", "generator_model": "GPT-5.6 Terra", "generated_at": "2026-09-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20260914-basic-counting-principle-0006", "batch_id": "batch-20260914-domain-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "93bab3d5cfdfbc202d67c4c86a274c346c203c51bc307262ef8fc7afb1adedf5", "family_id": "qf-basic-counting-principle", "seed": 20260999, "semantic_template": "qf-basic-counting-principle/{\"mode\":\"multiplication_fixed\",\"stages\":[3,3]}" }, "fields": { "title": "Boundary of the Basic Counting Principles", "stem": "A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 3 second-stage choices. How many combinations are there?", "answer": { "kind": "numeric", "value": 9, "explanation": "Every first-stage choice has a fixed 3 continuations, so 3×3=9." }, "skills": [ "basic counting principles", "condition recognition" ], "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": "一個流程有兩個連續階段；第一階段有 3 種選擇，且每一種之後第二階段固定有 3 種選擇。總組合數是多少？", "en": "A process has two successive stages. The first has 3 choices, and every first-stage choice is followed by exactly 3 second-stage choices. How many combinations are there?" } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [qf-basic-counting-principle, [branch_counts, conditional_stage_witness, mode, overlap_witness, stages]]", "after": "derived = [counting, null, null]", "relation": "implies", "rule_id": "R70_SOURCE_bank_specific_interface", "conditions": [], "explanation": { "zh-Hant": "只使用此bank真正明示的欄位、operation/domain與型別；BASE容許保留額外metadata，不把LDL SAFE/集合上限或其他bank的去重規則搬來。", "en": "Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "Counting fixed continuations per first-stage choice := mul(3, 3)", "after": "Counting fixed continuations per first-stage choice = 9", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "本步精確列出原操作數與運算；乘、減、除恰兩項。分數保持精確，衍生BigInt不重套raw SAFE，不用近似值湊答案。", "en": "This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Counting fixed continuations per first-stage choice，不以小數近似替代。", "en": "Recompute Counting fixed continuations per first-stage choice with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "3" ], "output": "9", "label": "Counting fixed continuations per first-stage choice" } }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "source roles = [3, 3]", "after": "derived = [9]", "relation": "implies", "rule_id": "Counting_FIXED_each_first_choice_has_same_second_count", "conditions": [], "explanation": { "zh-Hant": "每個第一階段選擇都恰有b個後續，依第一階段分成a個不重疊區塊，每塊b個，故a·b。這是固定後續數，不是隨機事件獨立性的宣稱。", "en": "EVERY first-stage choice has exactly b continuations.Partition by the a first choices into disjoint blocks of size b,so the total is a·b.This is a fixed continuation count,not probabilistic independence." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "反查本題原始操作數、角色與必要條件；重建下一步，不把單向推論倒當等價。數值/有限表核對不是任意命題的形式證明。原題、中英來源、MKO、AST、狀態與歷史均保留。", "en": "Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved." } }, "arithmetic": null } ], "math_model": { "result": 9 }, "conclusion": { "canonical_value": "9", "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-20260914-domain-foundations", "batch_path": "public/data/questions/batches/batch-20260914-domain-foundations/questions.jsonl", "batch_sha256": "726b3f94ccd05d8c91cc7e306b42fd71b6d9040e194db91739a250d488fd1de7", "original_line": 86, "original_line_sha256": "e6ffb0bb2b3721ef806cdeb274a76e31740757c02affe9976b4409a7335e8d3f", "question_compact_sha256": "93bab3d5cfdfbc202d67c4c86a274c346c203c51bc307262ef8fc7afb1adedf5", "english_row_compact_sha256": "e0a064f111374d3c01935d865a220f841d271ae830a5c0d305625157cf8b839b", "english_manifest_path": "public/data/i18n/editions/batch-20260914-domain-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20260914-domain-foundations.en.jsonl" }, "mko_bindings": { "mko-counting-principle": { "sha256": "0a0619ebbab98b96543b42abc4e66d927aeea705c65ee85b3a9badb6d3b3cf57", "english_sha256": "122425d1403554c17fc2a0e18e7b67e807bc50d412bad0250aa17710f6a34102", "original_path": "public/data/mko/counting-principle.json", "english_path": "public/data/i18n/editions/mko-counting-principle.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "20c37a7e64ee5969e7606c5cc3a00e575002ebebdfe82873521cc3d7c608caff", "original_candidate_sha256": "16b6d0b3a978859305778de9844bf1586dcaf8706cd7f40ce900ca4bf75c4d4d", "original_candidate_line_sha256": "16b6d0b3a978859305778de9844bf1586dcaf8706cd7f40ce900ca4bf75c4d4d" }, "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-20261026-012-p71", "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": "9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80", "question_first_witness_sha256": "c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "44e4e0f95c9634728da3e1b7a9a2d74012f6c5e595802a4d3cc56478c1db1084", "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." } }