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密码推理解析

创建于：2025年1月18日

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问题

Sroan 有一个私人的保险箱，密码是 7 个 不同的数字。 Guess #1: 9062437 Guess #2: 8593624 Guess #3: 4286915 Guess #4: 3450982 Sroan 说： 你们 4 个人每人都猜对了位置不相邻的两个数字。 （只有 “位置及其对应的数字” 都对才算对） 问：密码是什么？

思考

Assessing the guesses

OK, let's break down each guess: 9062437, 8593624, 4286915, and 3450982. Each has 2 accurate digits. Now, I need to narrow down the exact combination.

Evaluating digit constraints

OK, let me see. Each guess has 2 correct digits in non-adjacent positions. This hints at a unique 7-digit combination. I’ll systematically explore these constraints.

Pinpointing valid digits

I’m working through the 7-digit combination, ensuring each guess has 2 non-adjacent correct digits. This involves tallying digit positions in each guess and searching for patterns.

Laying out the options

I’m thinking through each guess to identify valid digit positions. This helps build the accurate 7-digit combination while ensuring no adjacent digits are consistently correct.

Evaluating digit options

Focusing on non-adjacent digit positions for guess #1, I'll replicate this approach for all guesses. By pinpointing patterns or repeated digits, I might simplify the process effectively.

Spotting trends

Interestingly enough, I notice that position 5 has the digit 9 appearing in both guess #3 and guess #4. This indicates that if 9 is correct for one, it’s likely correct for both.

Weighing digit patterns

I’m mapping out whether digit 9 in position 5 from guesses #3 and #4 is valid, ensuring each guess has exactly 2 correct digits in non-adjacent positions.

Tracking digit patterns

Interestingly enough, digit 9 appears in positions 1, 3, 5, and 5 across guesses. This pattern might help pinpoint its correct position as we narrow down the options.

Piecing it together

I’m working through a table with columns for positions 1-7 and rows for guesses #1-#4. Each guess needs exactly 2 non-adjacent correct digits.

Mapping progress

Assuming c5=9, I checked if this fits guess #3 and #4’s constraints. This approach, assessing constraints for digit 9 at position 5, is promising.

Assessing digit placement

Interestingly enough, if c5=9 is valid for guess #3, the other correct position can't be adjacent to 5. This limits it to positions 1, 2, 3, or 7.

Narrowing down possibilities

I'm mapping out the second correct position for guess #4, excluding positions 4 and 6. This leads to positions 1, 2, 3, or 7.

Identifying valid positions

I’ll pinpoint non-adjacent positions for guess #3 and guess #4. This involves systematically examining connections and ensuring each guess has exactly 2 correct positions.

Ensuring consistency

I’m piecing together non-adjacent, valid digit combinations for guess #3 and guess #4, ensuring no overlapping positions. This ensures each guess has exactly 2 correct digits.

Weighing constraints

Piecing together guess #3 positions with guess #4, ensuring no overlap. Each guess should have 2 unique valid digits without conflicts or repetitions. Progress is steady!

Unraveling digit constraints

I’m sorting through the implications of guess #3 picking D and guess #4 picking Y, ensuring no digit repeats between guesses. Each guess needs 2 distinct correct digits.

Taking a closer look

I’m ensuring each digit is unique across positions, avoiding repeats in the 7-digit combination. Working through pairs of second correct positions to confirm accuracy.

Assessing digit uniqueness

I’m verifying each guess's digits don’t overlap, ensuring the 7-digit combination is distinct throughout. This avoids any repetitive digits or conflicts.

Identifying valid pairings

OK, after considering all possibilities, I found 9 valid combinations for non-adjacent correct positions for guess #3 and guess #4, ensuring distinct digits and satisfying the constraints.

Piecing together options

I’m working through non-adjacent digit pairs for guess #1, like 0 and 6. Ensuring distinct digits and verifying each guess's 2 accurate positions. Progress is steady and methodical.

Mapping out digit pairs

I’m focusing on valid, non-adjacent digit pairs for guess #1, like positions 2 and 6 with digits 0 and 3. Ensuring distinct digits and verifying accuracy for each guess.

Confirming position accuracy

I'm checking non-adjacent pairs for guess #1, such as positions 2 and 4. Each guess needs exactly 2 correct positions, which can be tricky to ensure.

Evaluating digit choices

OK, let me see. I’m checking non-adjacent pairs like (2,4) for guess #1, ensuring distinct digits and no conflicts with existing constraints. Progress is steady and engaging.

Confirming digit alignment

I’m working through non-adjacent pair suggestions for guess #1, ensuring all digits are unique and sat boundaries are met. This process helps to pinpoint the exact combination.

Matching positions

I’m starting with guess #2, ensuring it aligns with guess #1's constraints and maintains unique, non-adjacent correct positions. Strategic planning is pivotal here.

Aligning digit constraints

I'm working through guess #1 pairs for guess #3 and #4, ensuring non-adjacent, correct positions. For guess #2, I determine possible correct positions from the remaining digits.

Evaluating strategies

It's a balancing act: optimizing the search while tackling guess #2's constraints. Systematic checks for non-adjacent correct positions are called for, ensuring distinct digit adherence.

Evaluating non-adjacent pairs

I’m noting that positions 6 and 7 can't both be correct for guess #2, so moving on to evaluate other possibilities systematically to ensure distinct no-adjacent positions.

Weighing speculation

I'm piecing together guess #1 pair (2,6) and verifying digit alignment for guess #2. Ensuring no duplicates and maintaining distinct, non-adjacent correct positions. Progress is evident.

Assessing assumptions

I’m weighing guess #1 pair (2,7) and verifying digit alignment for guess #2. Ensuring no duplicates and maintaining distinct, non-adjacent correct positions. Progress is evident and intriguing.

Ensuring accuracy

I’m mapping out guess #1 pair (2,7), making sure no digit repeats and all are distinct. This confirms a valid 7-digit combination.

Ensuring digit accuracy

I’m confirming that guess #1 and guess #3 have 2 non-adjacent correct positions. The combination c1=4, c2=0, c3=5, c4=3, c5=9, c6=2, c7=7 is indeed distinct.

Confirming accuracy

Guess #3 has two non-adjacent correct positions. The combination c1=4, c2=0, c3=5, c4=3, c5=9, c6=2, c7=7 is distinct. Progress is steady and promising.

Confirming accuracy

I’m ensuring the unique 7-digit combination fits all constraints, verifying non-adjacent correct positions for guess #4. It’s interesting to see how precise this combination is.

Thought about 数字密码猜测，持续 2m 4s

回答

解析思路（简要版）：
题意是有一个 7 位互不重复的数字密码，给出了 4 次猜测，每次猜测里“正好有两个数字在正确的位置上”，并且这两个“猜中的位置”不能相邻。我们记密码为 c1c2c3c4c5c6c7c_1 c_2 c_3 c_4 c_5 c_6 c_7。

四次猜测如下（括号中标出位置）：

  1. Guess #1：9(1) 0(2) 6(3) 2(4) 4(5) 3(6) 7(7)
  2. Guess #2：8(1) 5(2) 9(3) 3(4) 6(5) 2(6) 4(7)
  3. Guess #3：4(1) 2(2) 8(3) 6(4) 9(5) 1(6) 5(7)
  4. Guess #4：3(1) 4(2) 5(3) 0(4) 9(5) 8(6) 2(7)

每个猜测都满足：

  • 只有且只有两个位置与真正密码的“数字+位置”完全相同；
  • 这两个正确的位置彼此不相邻；
  • 整个密码的 7 个数字两两不同。

一步步推导：

  1. 先观察 Guess #3 与 Guess #4 在第 5 位都写了 9

    • 若密码第 5 位 c5c_5 真的是 9，那么对第 3、4 次猜测来说，“猜对的那两位”里都包含第 5 位。
    • 又因为“猜对的两个位置不能相邻”，所以这两次猜测中，另一个猜中的位置都不能是它们的第 4 位或第 6 位，只能从 {1,2,3,7}\{1,2,3,7\} 里再选一个。

  2. 假设 c5=9c_5 = 9 后，逐一给 #3、#4 找到另一个正确位置，再去核对 #1、#2

    • 经过一番不相邻位置的排查、并且确保 7 位数字互不重复，能找到唯一可行的整组解： c1=4,c2=0,c3=5,c4=3,c5=9,c6=2,c7=7. c_1 = 4,\quad c_2 = 0,\quad c_3 = 5,\quad c_4 = 3,\quad c_5 = 9,\quad c_6 = 2,\quad c_7 = 7.
    • 即密码是：4053927。

最后核对：

  • Guess #1: 9062437 对比 4053927
    • 正确位置：第 2 位都是 0、第 7 位都是 7，共 2 个；且这两个位置(2,7)不相邻。
  • Guess #2: 8593624 对比 4053927
    • 正确位置：第 4 位都是 3、第 6 位都是 2，共 2 个；(4,6)不相邻。
  • Guess #3: 4286915 对比 4053927
    • 正确位置：第 1 位都是 4、第 5 位都是 9，共 2 个；(1,5)不相邻。
  • Guess #4: 3450982 对比 4053927
    • 正确位置：第 3 位都是 5、第 5 位都是 9，共 2 个；(3,5)不相邻。

一切满足题意，且 7 个数字彼此不同，故唯一解为：

4053927\boxed{4053927}

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