GK 2020 Q25

Let’s call the four-digit number \(N\). The last clue tells us that none of the digits 7, 6, 4 or 2 is a digit in \(N\). Then the fourth clue shows that 1 is a digit in \(N\), but it is not the fourth digit. The first clue now tells us that \(N\) involves a 3 but not as its third digit. It also shows that 1 is not the second digit. The second clue now tells us that either 8 is the second digit of \(N\) and 9 is not one of its digits or else 9 is the first digit of \(N\) and 8 is not one of its digits. Suppose that 8 were the correct second digit. Then the third clue would tell us that both 0 and 5 were correct digits. But this would mean that all of 1, 3, 8, 0 and 5 were digits of the four-digit number \(N\). Therefore 8 is incorrect and so 9 is correct and is the first digit. Knowing this, the third clue shows us that exactly one of 5 and 0 is correct and, moreover, it is in the right place. It can’t be 5 because the first digit of \(N\) is 9. So 0 is the correct second digit. We already know 3 is correct, but is not the third digit; so the last digit of \(N\) is 3 and \(N\) is 9013.