I took the practical approach and came up with a strategy that could be optimal (I have not tried to prove it).
In the worst case scenario it takes:
I made a drawing to show my strategy. The yellow squares are the rooms the police officer looks into that night, the black squares are rooms in which the thief logically cannot be.
I managed to find the 8 in
So we are trying to find the 5th largest card in a bunch of 12, by measuring them in batches of four. Here's my strategy:
Now we have identified, for certain, a couple of cards we can exclude: A1 and A2 (both have at least 8 cards smaller than them), and B4, C2, C3 and C4 (all have at least 5 cards bigger than them). We also have ...
I can do it in
Assume the worst case. Any other results would obviously be easier to solve.
For any of these options:
Thus, we have completed the process and identified card 8 in at most
Note, that as a side-effect of identifying card 8, we also identified cards 12, 11, 10, & 9. There might be a more efficient option to ...