The Youtube channel MindYourDecisions posted a video about the "Hiding Cat Puzzle". In the puzzle, you have a line of five boxes, one of which contains a cat. You don't know where the cat is, and your goal is to find it. You can look in any box, but each time you do so, the cat moves to an adjacent box without your knowledge. Interestingly, it is possible to find the cat by strategically choosing which boxes to look in. You can watch the video for the solution. I found this puzzle fascinating, and thought about how it might generalize to a grid of boxes. So here is my riddle for you:
You are presented a $10\times 10$ grid of boxes, one of which contains a cat. You may select $n$ boxes to look in, and then the cat moves to an adjacent box without your knowledge. Two boxes are considered adjacent if they touch vertically or horizontally, but not diagonally. After the cat moves, you may choose another $n$ boxes to look in, and then the cat moves again, etc. The question is, what is the lowest number $n$ such that you can be guaranteed to find the cat? And give an example of a successful strategy.