Solve 3 digit number lock puzzle 370 in two steps from 5 clues, each showing a 3 digit hint code with a statement on digit and positional correctness.
3 Digit Number Lock Puzzle Including Hint Code 370
Following are the five clues:
Clue 1: Code 3 7 0: Two digits are correct but both are in the wrong places.
Clue 2: Code 2 5 9: One digit is correct but in the wrong place.
Clue 3: Code 9 3 5: Two digits are correct and in the right places.
Clue 4: Code 9 5 3: Two digits are correct — 1 in the right place and 1 in the wrong place.
Clue 5: Code 1 6 4: No digit is correct.
Find the 3 digit code to open the number lock using a short two-step logical analysis.
Time to solve: 10 minutes.
Solution to Solve 3 Digit Number Lock Puzzle 370 in Two Steps
Clue 5 eliminates digits 1, 6 and 4, but none of these appear anywhere in the hint codes of other four clues. Use positional and cross-clue logic to deduce the three correct digits.
Step 1. False assumption logic and positional correctness analysis confirm two correct digits
Use most promising Clues 2, 3 and 4 together. Reasons for the choice:
- First key pattern: all three are positive clues with positive digit correctness and wrong/right position descriptions.
- Second key pattern: digit 9 is common in all three clues. This makes the group of clues most promising for applying false assumption logic on digit 9 and get the first breakthrough.
Clue 2: Code 2 5 9: One digit is correct but in the wrong place.
Clue 3: Code 9 3 5: Two digits are correct and in the right places.
Clue 4: Code 9 5 3: Two digits are correct — one in the right place and one in the wrong place.
- False assumption on Clues 2+3+4: In Clues 3 and 4, assume 9 wrong ➡ two correct digits 3 in middle and 5 rightmost in Clue 3 violate position statements for 3 and 5 in Clue 4 ➡ Contradiction because of the assumption ➡ Assumption is wrong.
- ➡ Conclusion: One of 3 and 5 is right and 9 right in leftmost.
- So we have partial code: [ 9 ? ? ].
- Now combine with Clue 2 (2 5 9): "One digit correct but in the wrong place." Since 9 is already identified as correct, both 2 and 5 must be incorrect.
- Consider again Clue 3 (9 3 5): "Two correct digits in right places." As 9 is correct and 5 incorrect, digit 3 must be the second correct digit in middle.
- Partial code becomes [ 9 3 ? ].
Step 2. Digit lockout determines the third digit
The search for the third correct digit must be in Clue 1 because all digits in hint codes of three clues analyzed are known - a new digit is needed — and Clue 1 has two probable candidates for the third correct digit — 0 and 7.
Clue 1: Code 3 7 0: Two digits are correct but both are in the wrong places.
- Clue 1 (3 7 0): We already have 9 leftmost and 3 in the middle in the lock opening code. These two positions are blocked. If 0 were the right digit, being stated in wrong place in the clue, it has no other position to occupy in the lock opening code — it suffers the fate of digit lockout and must be wrong.
- That leaves digit 7 as the second correct digit in the clue and its position is rightmost. This satisfies the "wrong place" statement of the second correct digit as well.
- Solution: Lock opening code: [9 3 7].
Verification
Check all clues against correct code 937:
- Clue 1 (3 7 0): Two digits are correct but both are in the wrong places: Digits 3 and 7 are present but both in wrong places — matches.
- Clue 2 (2 5 9): One digit is correct but in the wrong place: Digit 9 is present but shown in wrong place (third) — matches.
- Clue 3 (9 3 5): Two digits are correct and in the right places: Digits 9 and 3 are correct and in the right places — matches.
- Clue 4 (9 5 3): Two digits are correct — 1 in the right place and 1 in the wrong place: Two digits present — 9 in right place (left), 3 in wrong place in the clue — matches.
- Clue 5 (1 6 4): No digit is correct: None of 1,6,4 in correct code 937 and so are incorrect — matches.
Answer: 937
Lessons learned
1. Detection of key patterns identifies three clues to analyze in combination for breakthrough. The key patterns are based on a digit common to multiple clues:
- Pattern 1: All three are positive Clues 2, 3 and 4 with positive digit correctness and wrong/right position descriptions.
- Pattern 2: Digit 9 common to all three clues.
- The patterns identify the clues most promising for combined analysis, and
- Ideal for applying false assumption logic on digit 9 to get the first breakthrough.
2. False assumption logic — Use of key patterns: Assume the common digit wrong. In case of a contradiction, the false assumption is proved wrong, and the wrong digit right. Otherwise, no contradiction (in rare cases) implies (does not ascertain) validity of the assumption in the clues analyzed. Verification from remaining clues would then create the breakthrough.
- Digit 9 assumed wrong led to a contradiction in Clues 3 and 4 in conflicting position and correctness descriptions. It proved the assumption of digit 9 right in its leftmost position in the lock opening code.
3. Leveraging the correct/incorrect digit information:
- From Clue 2, digit 9 correct proved digits 2 and 5 incorrect. This confirmed from Clue 3, the second correct digit and its place in middle. Partial code formed: [ 9 3 ? ].
- The double breakthrough highlights the advantage of combined analysis of multiple clues based on key patterns.
4. Importance of forming the partial code:
- Partial code [ 9 3 ? ] revealed: Two places occupied in the lock opening code with only the rightmost still vacant for the third correct digit.
5. Choice of the Clue for next breakthrough: The only remaining clue containing probable third correct digit to be chosen, as analytical breakthrough potential of all other clues are exhausted. All digits in three clues analyzed are identified as correct or incorrect.
- Clue 3 chosen accordingly for next breakthrough. It promised one of 0 or 7 to be the third correct digit.
6. Digit lockout: By position description of a digit in a clue, the digit suffered from digit lockout with no place in the correct code. This confirmed the digit incorrect.
- In Clue 1, digit 0 in rightmost position is locked out because, the clue stated the only correct digit in its hint code [3 7 0 ] to be wrongly placed — with leftmost and middle position occupied by 9 and 3, digit 0 has no valid place in the code and proved incorrect.
- With 0 incorrect, second candidate 7 automatically confirmed correct in rightmost position.
7. All in successive pattern based right clue choice, using the patterns for position and digit correctness analysis and successive breakthroughs in steps.
Solution reached in two confident steps.
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