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.

Look at the 2nd and 5th attempt. What can you deduce?

… you should be able to deduce one number, but not its position.

The first attempt should now tell you more.

Which number in the third attempt is correct and in the correct place?

The fourth attempt should now seal it.

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 $042.$