# Find the burglar by covering all the streets

There is a burglar in a neighborhood and the police officers are trying to find him. But there are limited amount of police, namely 4 of them, who could cover the streets.

Could you place our police officers in the neighborhood below to guarantee to find our burglar wherever he might pop up?

Source: 15 August 1995, Turkish IVth Mind games Championship Qualifying Exam

• i suppose you are looking for more of an answer than "yes, you can" :) May 12 '20 at 21:01
• @SteveV hahah :) yes
– Oray
May 12 '20 at 21:06

I believe this should work! Large circles are policemen places. I’ve numbered them 1,2,3,4, and then each line segment in the diagram has a 1,2,3,4 corresponding to which officer sees that segment.

To add some reasoning on how to get to the solution:

In order to cover the 4 streets on the outside, we must place at least 2 policemen in the outer corners, opposite to each other. There are only 2 ways of doing so. That leaves only 2 policemen that need to be placed, and it is relatively easy to find where they need to be placed, even by just trying all possible combinations.

Another way to get to the solution, no guessing needed. No spoilers as two solutions were posted already.

There are 13 lines. This means one officer needs to cover 4 lines and remaining 3 policemen need to cover 3 lines each, assuming no overlap. There are 3 places with 4 lines (and none with more than 4): top left, mid left and the mid near bottom. All three places see each other - if we place officers on two of them, they together still cover just 7 places (as if one spot had 3 lines), meaning it is not possible to place an officer on a spot overseeing just 2 lines or a spot with 3 lines seeing one of our policemen.

This already dramatically decreases number of possible spots for our officers. We could start guessing, but we can do better.

Observe bottom line. Out of theoretically 5 spots for policemen we are down to just 2 corners with 3 lines each, other spots have just 2 lines and can't lead to any solution. Bottom left corner sees all places covering 4 lines. This would mean as if it had 2 unique lines to add. Meaning the bottom right corner has a policeman as it is the only one remaining.

That policeman sees 2 spots with 4 lines. Those are eliminated, meaning another policeman is in the top left corner. Just 2 spots with 3 lines remain - put the remaining two policemen there for our unique solution.