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密碼推理解答

Created on: January 19, 2025

Answered using O1 by Chat01

Question

Sroan 的私人保險箱密碼為 6 個不同的數字。現有三人猜測密碼，A 猜 891735，B 猜 517384，C 猜 721846。
Sroan 說："你們每個人都猜對了兩個數字，且這兩個數字的位置不相鄰。"
假設三人足夠聰明，請問他們能否推理出正確的密碼？密碼是多少？

Thought

Mapping out correct digits

I'm analyzing the puzzle to identify the correct code, noting each guess has exactly 2 accurate digits in non-adjacent positions. This ensures a unique solution.

Ensuring distinct digits

I’m pulling together a plan to ensure no duplicate digits and identify the correct non-adjacent digits for each guess, leading to a unique 6-digit code.

Isolating digit patterns

I’m isolating digits from each guess to identify patterns. Repeated digits in consistent positions across guesses could be critical clues, helping pinpoint the correct code positions.

Spotting digit repeats

Noticing digit "1" appears twice in position 3, reveals interesting potential for it to be correct in both guesses A and C, pointing towards a possible solution.

Piecing together correct positions

OK, I’m thinking each digit's position is right once among the guesses. This method ensures two accurate digits for each guess while keeping non-adjacent constraints intact.

Mapping digits

I'm working through each digit's positions to ensure no duplicates and track their correct alignment across guesses, assuming each position is accurately guessed once.

Piecing together matrix

I'm crafting a matrix with guesses as columns, aiming to pinpoint one accurate digit per column, while ensuring each guess has two correct digits in non-adjacent positions.

Mapping non-adjacent selections

I’m identifying possible ways to partition 6 positions into 3 non-adjacent pairs for guesses A, B, and C, ensuring each has 2 correct digits. This leads to viable code configurations.

Ensuring completeness

I’m covering all positions with three distinct non-adjacent pairs, ensuring each guess has two correct digits. This method guarantees a thorough and systematic approach.

Examining partition options

Analyzing different partitions to ensure each guess has two correct digits in non-adjacent positions, I find the valid partition: (1,3), (2,5), (4,6).

Piecing together valid partitions

I’m pinpointing partitions with non-adjacent pairs for guesses A, B, and C. Valid partitions include (1,4), (2,5), (3,6) and (1,5), (2,4), (3,6). Progress is evident.

Finalizing partitions

I’m working through partitions with non-adjacent pairs for guesses A, B, and C. Valid partitions identified include (1,6), (2,4), (3,5) and (1,6), (2,5), (3,4).

Mapping the partition puzzle

I’m piecing together five partitions for guesses A, B, and C, each with non-adjacent pairs. The goal is to assign these pairs consistently to validate each guess's constraints.

Assigning pairs

I’m systematically assigning pairs from each partition to guesses A, B, and C, ensuring each guess has two correct digits in non-adjacent positions, and each digit is distinct.

Trying new ways

I’m testing different pair assignments to A, B, and C. Ensuring unique digits for each code position, I’m verifying if any configuration meets the constraints without duplication or conflicts.

Ensuring digit uniqueness

Cross-referencing permutations to confirm no repeating digits. The valid code sequence found is 5 9 7 8 3 6.

Confirming accuracy

I’m cross-checking the valid code sequence against guesses A and B. Each guess aligns with two correct, non-adjacent digits in the specific positions, ensuring consistency and accuracy.

Confirming alignment

I’m verifying the valid code sequence against guesses A, B, and C. Each guess has two correct digits in non-adjacent positions, ensuring the code fits all constraints.

Pulled together permutations

I’m pulling together permutations of partition #1 to spot the correct code. Progress is evident in matching constraints and ensuring each guess has two non-adjacent correct digits.

Assessing permutation validity

I’m testing various permutations to ensure no digit repeats and each guess has two correct digits in non-adjacent positions. Only one permutation meets all criteria.

Testing permutations

I’m digging into permutations of partition #2, ensuring no repeating digits and meeting all constraints. Each guess must have two non-adjacent, correct digits.

Trying permutations

I’m testing permutations of partition #3, ensuring no repeated digits and all constraints are met. Each guess needs two non-adjacent, correct digits. Progress is becoming clearer.

Testing partition permutations

I’m trying permutations of partition #4 to ensure no digit repetitions and validate all constraints. Each guess should have two non-adjacent, correct digits.

Ensuring code validity

I’m piecing together permutations of partition #4 to confirm no digit repetitions and validate constraints. The valid code sequence 5 9 1 7 8 6 checks out for all guesses.

Pinpointing errors

I’m examining the valid code sequence against guesses A, B, and C. Each guess has two correct digits in non-adjacent positions, but I’m realizing that the solution still needs refinement.

Evaluating permutations

I’m examining permutations of partition #5 to ensure no digit repetitions and validate all constraints. Progress is evident in meeting the requirement of two non-adjacent, correct digits for each guess.

Testing configurations

I’m trying permutations of partition #5 to ensure no digit repetitions and validate constraints. The code 5 2 1 8 3 4 meets all requirements.

Confirming accuracy

I’m mapping out the code sequence against guesses A, B, and C. Each guess aligns with two correct digits in non-adjacent positions, meeting all constraints and ensuring a unique solution.

Finalizing permutations

I’m working through permutations of partition #6, ensuring no digit repeats and all constraints are met. Each guess should have two non-adjacent, correct digits for a valid code sequence.

Thought about密码推理게임，持续 1m 13s

Answer

以下是一種較系統的解法示例，說明為何唯一能推出的密碼是 597836。

題目整理

  • 私人保險箱密碼為 6 個「互不重複」的數字。

  • 三人的猜測如下：
    A：8 9 1 7 3 5
    B：5 1 7 3 8 4
    C：7 2 1 8 4 6

  • Sroan 提示：
    「每個人都 恰好 猜對了 2 個 數字，且這 2 個正確數字在密碼中的位置 互不相鄰。」

目標：找出唯一的正確密碼。

核心思路

  1. 每位猜測者都剛好對 2 碼、且這 2 碼的位置不相鄰。
  2. 對於 6 碼密碼，如果用 A、B、C 三次猜測，都各對 2 碼，則「總共 2+2+2 = 6 個正確位置」。很常見的情形是：在每個位置上，正好只有一個人的猜測是對的。
  3. 我們可以把「密碼的第 1～6 位」想成要分配給 A、B、C 的「正確位置」。由於每人正好 2 個位置對，且那 2 個位置不可相鄰，就等於要把 6 個位置分成三組 {(p1,p2)},{(q1,q2)},{(r1,r2)}\{(p_1,p_2)\}, \{(q_1,q_2)\}, \{(r_1,r_2)\}，分別給 A、B、C，並確保每組內的兩個位置不相鄰。

系統性檢驗

第一步：列出「兩兩不相鄰」的可能配對

6 個位置（1,2,3,4,5,6）中，「可組成不相鄰的二元組」包含下列幾種：

text
(1,3), (1,4), (1,5), (1,6),
(2,4), (2,5), (2,6),
(3,5), (3,6),
(4,6).

要分成三對、覆蓋 1~6，每個位置只能出現一次，而且每對之內不能是相鄰數字。經過系統列舉，能把 {1,2,3,4,5,6} 完整且不重疊、不相鄰地分成三對的方式大約有下列 5 種「三對分組」(不考慮組內順序)，例如：

  1. (1,3), (2,5), (4,6)
  2. (1,4), (2,5), (3,6)
  3. (1,4), (2,6), (3,5)
  4. (1,5), (2,4), (3,6)
  5. (1,6), (2,4), (3,5)

接下來，每一種「三對分組」還要再分別指派給 A、B、C。由於 A、B、C 各自要有 exactly 2 個正確位置，所以等於要把這三對 分配 給 A、B、C（即一個對應一組）。

第二步：帶入具體數字檢查

以第一種分組 (1,3), (2,5), (4,6) 為例，嘗試把它分派給 A、B、C 的所有可能，看看最終能否得到「6 個不重複的數字」且「每位猜測者都只對那 2 個位置」的情況。

  • 假設

    • A 對 (2,5)
    • B 對 (1,3)
    • C 對 (4,6)

    代表：

    1. 密碼第 2、5 位 = A 的第 2、5 位
      • A 猜的是 8 9 1 7 3 5
      • 所以密碼第 2 位 = 9、密碼第 5 位 = 3
    2. 密碼第 1、3 位 = B 的第 1、3 位
      • B 猜的是 5 1 7 3 8 4
      • 所以密碼第 1 位 = 5、密碼第 3 位 = 7
    3. 密碼第 4、6 位 = C 的第 4、6 位
      • C 猜的是 7 2 1 8 4 6
      • 所以密碼第 4 位 = 8、密碼第 6 位 = 6

    如此便得到候選密碼：

    D1D2D3D4D5D6=5 9 7 8 3 6. D_1 D_2 D_3 D_4 D_5 D_6 = 5\,9\,7\,8\,3\,6.
    • 檢查是否有數字重複？
      5、9、7、8、3、6 全都不同。
    • 檢查對 A 而言，是否只對了第 2、5 位？
      • A 猜：8 9 1 7 3 5
      • 密碼：5 9 7 8 3 6
        • 第 2 位 9（相同）
        • 第 5 位 3（相同）
        • 其餘 1、4、6 位與 A 的猜測都不同，正好只對 2 處，且 2、5 位置不相鄰，符合！
    • 同理檢查 B、C，也皆僅對應分配的那 2 個位置；也都滿足「不相鄰」的條件。

進一步把其他分組、其他指派方式都測一遍，會發現只有這種分配方式產生了合法且唯一的密碼：

 5 9 7 8 3 6 .\boxed{\,5\,9\,7\,8\,3\,6\,}.

結論

在符合「三人各對 2 碼、位置不相鄰、且密碼 6 碼互不重複」的條件下，唯一能推出的正確密碼便是

597836.\boxed{597836}.

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