Caves and A Thief Puzzle

February 2, 2022

Puzzle on A Thief and Caves

The puzzle is based on a situation where we have caves and a thief puzzle. Let us look at the premise, rule and objective to find out more.

Premise – There are 13 caves arranged in a circle. In one of the caves, there is a thief. Each day, the thief has the option of moving to any of the nearby caverns or stay in the cave where he was the day before.
Rules – Cops are also allowed to enter any two caves they want each day.
Objective – What is the minimal number of days required to ensure that the thief is apprehended by the cops?

Note – The thief may or may not move to a nearby cave. Cops can inspect any two caverns, which do not have to be contiguous.

Solution to Find Minimum Number of Days

Let us assume the burglar is in cave C1 and is working his way clockwise.

The cops begin their hunt from caves C13 and C12 on the first day.
On the second day, visit Cave C13 and C11, then C13 and C10 on the third day, and so on until they reach Cave C13 and C1 on the 12th day.
So the goal is to check C13 every day so that if a thief tries to move anti-clockwise, you can catch him right away, and if he goes clockwise, cops can capture him in no more than 12 days (this include the case where he remains in Cave C1).
Therefore the correct answer is 12.

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