Here is a question I came up with and i have been pondering for a while. It relates to covering codes, a subset of coding theory. I could not come up with an adequate solution, so here I am, asking the forum!
Suppose you and I are playing a game. I think of a 5-digit binary number, and you have a certain number of guesses to guess what it is. You write all your guesses on a piece of paper, and when you have guessed to your hearts contet, you show me the slip of paper. I tell you if any of your guesses were correct. The catch is, your guess can be 1 digit off and still be considered correct. For example, if k=5 and I am thinking of 11101, you could guess either 11101, 11100, 11111, 11001, 10101, or 01101. My question is, what is the least number of guesses you need to write down to make absolutely sure you have properly guessed my number with one of your guesses?
Now I have done some research and found that the answer is 7. In other words, you need to guess 7 numbers to make sure you have guessed my number. But I cannot find what these 7 numbers are! As such, I refuse to simply accept this answer. Furthermore, the answer could be 6 (since each number guess covers 6 possibilities, the absolute minimum number of guesses you need is 6 since 6x6=36>32). But obviously 6 is not the answer... is there a logical reason why?