 # Practice Paper 3 Question 3

A 3 digit lock gives feedback when trying out a combination. What is the correct unlocking combination if the lock responds as follows for the following attempts:

• $$\tt206:$$ two numbers are correct but wrongly placed
• $$\tt738:$$ no numbers are correct
• $$\tt682:$$ one number is correct and correctly placed
• $$\tt614:$$ one number is correct but wrongly placed
• $$\tt780:$$ one number is correct but wrongly placed

Briefly justify your answer.

## Hints

• Hint 1
Look at the 2nd and 5th attempt. What can you deduce?
• Hint 2
... you should be able to deduce one number, but not its position.
• Hint 3
The first attempt should now tell you more.
• Hint 4
Which number in the third attempt is correct and in the correct place?
• Hint 5
The fourth attempt should now seal it.

## Solution

Using the second and fifth attempts, we can deduce that $$0$$ is a correct digit. Furthermore, we can conclude it is the first digit using the first and fifth attempts.

We can say that either $$2$$ or $$8$$ is in the correct place in the third attempt as we know that $$0$$ is the first digit. Then using the second attempt, we conclude that $$2$$ is the last digit. Note that this also tells us $$6$$ is incorrect.

As the correct number in the fourth attempt must be the second digit and is wrongly placed, it must be either $$4$$ or $$6.$$ However, we know that $$6$$ is incorrect. Therefore the correct unlocking combination is $$042.$$

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