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.
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Look at the 2nd and 5th attempt. What can you deduce?
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… you should be able to deduce one number, but not its position.
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The first attempt should now tell you more.
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Which number in the third attempt is correct and in the correct place?
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The fourth attempt should now seal it.
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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$042.$