These silly riddles are not fit for a chess forum. Now try this:
Ghost hunting.
A somewhat peculiar complex of 16 houses in a row is being haunted by a ghost. All the inhabitants have been scared off and the houses are now empty. The ghost, you have been told, appears one time during the night in just one house, and reappears the following night in one of the adjacent houses. It is never seen by day.
You are the ghost hunter. OK, you can be the ghost buster if that makes you happy. You are capable of setting up your stuff inside just one house per night, effectively catching and/or driving away the ghost if it happens to appear in that house. The problem is, you can't see the ghost unless you're inside the right house, so you have no idea in which house it's currently located.
The ghost is far from harmless, and none of the townsfolk seem intent on helping you out. They're hiding out in the tavern waiting for you to resolve the situation. It's the first of february and the Mayor has promised to pay you double your normal charge if you can have this fixed by the end of the month.
How can you catch the ghost in just 28 nights?
Edit: Sorry, I got the number wrong first, now it says 16 houses which is how it should be. Also, I forgot to specify that it's not safe for you to search houses unless you've set up your stuff and taken special ghost hunter measures during the day. Meaning, obviously, that you can only enter one house per night.
It looks like the idea here is to figure out if you are an even number of houses away or not as soon as possible. I think you can determine that within the first 14 nights. Once you determined you are an even number of houses away from the ghost, then you can use the next 14 nights to catch him by going next door every night. Lets say you have houses #1-#16. I'm assuming the worst case scenario.
Stay in house #2 for the first night. Move next door to the right to house #3 for the next night, house #4 for the 3rd night, etc... After you reach house #15 on the 14th night, then you either have caught the ghost, or the ghost is an odd number of houses away from you. If he was an even number of houses away to start with, then he remains an even number of houses away through out the whole sequence and should get caught by the 14th night. This would also mean that the ghost was not originally in house #1 on the first night since house #1 is just one house away from house #2. Let A = your house # and B = the ghost's house #. If B - A = even, then (B +/- 1) - (A + 1) = even. The symbol "+/-" means plus or minus. Also, if B - A = odd, then (B +/- 1) - (A + 1) = odd.
If you have not caught him yet, then stay in house #15 for the 15th night. By staying in the same house, you should now be an even number of houses away from the ghost (odd +/- 1 = even). Move next door to the left to house #14 for the 16th night. Keep moving next door to the left until you spent your last night (28th night) in house #2.
Can the ghost bypass you like if you are at house 2 but next night you go to house 3 can he be at house 2?