Is there a winning strategy in this game? There are 8 smugglers standing in a row and they are waiting for customs inspecting. One of the smugglers takes a small bag which contains smugglings. There is one officer who can examine one smuggler at a time. After every examination, smuggler with the bag must pass the bag to another smuggler who stands next to him. The officer may examine as many times as he wants till he finds the bag.
The question：
Is there a winning strategy for officer? What tool should I use to solve this question?
 A: Label the smugglers 1, 2, 3, ..., 8, and make two passes.
The first pass goes 2, 3, 4, 5, 6, 7. If the contraband started on an even-valued smuggler it must be discovered in this pass. If not, it started on an odd-valued smuggler and the second pass is just counting back down in the reverse of the same pattern.
ETA:
Assume it starts on an even number, i.e. it is initially on one of 2, 4, 6, or 8. 


*

*Check 2. If we didn't find it, then it must have started on 4, 6, or 8, so now it can only be at 3, 5, or 7. 

*Check 3. If we didn't find it, the it must have been on 5 or 7, so now it has been passed to 4, 6, or 8. 

*Check 4. If we didn't find it, then it must have been on 6 or 8, so now it has been passed to 5 or 7. 

*Check 5. If we didn't find it, then it must have been on 7, so now it has been passed to 6 or 8.

*Check 6. If we didn't find it, then it must have been on 8, so it has been passed to 7.

*Check 7. If we didn't find it, it must have started on an odd value, so move on to the second pass.

