The number lock is 394.
Initially, we can’t draw any conclusions from Clue 1 alone. However, combining Clue 1 and Clue 2 gives us some insight.
Analysis: Since digit 2 cannot be in the same leftmost position in both 291 and 245 while being the correct digit in its rightful place in 291 and the right digit in the wrong place in 245, this leads to a contradiction.
Conclusion 1: Therefore, digit 2 is incorrect and not part of the code.
Next, considering Clue 4, which states that all digits in 578 are incorrect, we can conclude:
Conclusion 2: Since both digits 2 and 5 are ruled out from clue 245, digit 4 must be correct and can only occupy either the leftmost or rightmost position.
Now, we see that digit 4 appears again in Clue 3 (463), which mentions that two digits are correct but both are misplaced. From this, we deduce:
Conclusion 3: As digit 4 is ruled out in the leftmost position in 463 and the middle position in 245, the correct position for digit 4 must be at the rightmost.
So far, we have the code format: “? ? 4”, with the first two digits still unknown.
Let’s revisit Clue 1:
Clue 1: Code 291: One digit is correct and in the correct position.
From our previous deductions, we know:
Digit 2 is incorrect.
The rightmost position is reserved for digit 4.
Thus,
Conclusion 4: Digit 1 in 291 must be incorrect (due to the conflict with digit 4’s position), leaving digit 9 as the only possibility in the correct middle position.
This is a significant breakthrough. The code format is now: “? 9 4”, with only the leftmost digit yet to be identified.
Looking at Clue 5, which states that in code 569, one digit is correct and in the wrong position, we find that:
Conclusion 5: Since 9 is indeed the correct digit and 5 is ruled out, both digit 6 and digit 5 must be incorrect.
Lastly, Clue 3 states that two digits in 463 are correct but misplaced. This gives us crucial information:
Conclusion 6: Given that digit 6 is ruled out and digit 4 is confirmed as correct, digit 3 must be the second correct digit, filling the vacant leftmost position in the code.
Thus, the code to unlock the number lock is 394.