A Magic Trick
Pete and Zandra are performing a mathematical magic trick. Given a string of \(N\) digits (0-9) of an audience member's choice that only Pete can see, Pete will cover up a digit of his choosing, and then Zandra will look at the remaining digits (and which one is covered) and -- without any communication with Pete -- say the digit that Pete covered.
For example, if \(N=13,\) the audience member might write 2, 5, 1, 1, 5, 6, 2, 3, 3, 7, 8, 9, 2. Then, Pete decides to cover up the 7. Then, Zandra opens her eyes, looks at the whole list (but can't see the covered number), and then 'guesses' the missing number, 7, correctly. Since that number might have been any digit, this seems like a pretty amazing feat.
What is the smallest \(N\) for which this trick is always possible? And how could they pull it off?
Bonus: What if Pete had to cover two adjacent digits?