Earlier today I set you the following puzzle.
Police chase
The streets of the city are a square grid that extends infinitely in all directions. One of the streets, Broadway, has a police offer stationed every 100 blocks.
A robber is somewhere in the city.
Can you devise a strategy that guarantees the criminal will be spotted by the police at some point in time?
Relevant information: The robber and the police officers are in a street at all times. The robber has a finite maximum running speed, which is faster than any officer’s. The police can see infinitely far.
Solution
You need to devise a system with two elements: first, in which officers are placed on every intersection on Broadway. If this is the case, officers will be able to see down every perpendicular, meaning that the robber cannot escape down any perpendicular. Second, officers need to walk down all the roads perpendicular to Broadway, which enables them to look down all the streets that are parallel to Broadway. If this is the case, all streets are covered and the robber has nowhere to hide.
First, some terminology. Let’s call the roads parallel to Broadway “avenues”, and those perpendicular to Broadway “streets”. And lets number the streets from –infinity to +infinity.
We begin with police officers stationed on Broadway at streets 0, +100, -100, 200, -200 etc.
Step 1. At the moment the search is announced, let the officers on the ‘odd hundreds’ (i.e. +/-100, +/-300, +/- 500…) start walking towards Street 0. And let the officers on the ‘even hundreds’ (i.e. 0, +/-200, +/-400, …) stay put.
Wherever the robber is, they are now constrained by a section of the city that is 200 blocks wide, since whenever they cross a street that is a multiple of 200 they will be seen by the officer at the intersection of that street and Broadway.
Step 2. Let the officers from the ‘odd hundreds’, i.e. the streets numbered +/-100, +/-300, +/- 500,…, stop walking down Broadway when they reach Street +/-1, Street +/-2, +/-3, ..and then tell them to turn right, and carry on walking. The officers from the + streets will walk one way, and the officers from the – streets will walk in the opposite direction. As each of them walks down their respective streets they will be able to look down every avenue as they pass them.
Since there are an infinite number of police officers, there will eventually be officers walking down streets +/-1, +/-2, +/-3.. +/-n for all finite n. In other words, at some point all the streets in the 200 block where the robber is will have officers in them. In order for the robber not to be caught, the robber must hide in an avenue. Sooner or later, as the officers move away from Broadway, checking every avenue as they pass it, the robber will be caught.
I hope you enjoyed today’s puzzle. I’ll be back in two weeks.
Thanks to Professor Alex Lvovsky of the University of Oxford for this puzzle. Prof Lvovsky is the head of COMPOS, an online program that gives free tuition in maths and physics for students in years 10, 11 and 12 (GCSE and A-level). The idea is to enable enthusiastic teenagers to learn these subjects at a deep level, with regular tiutorials by Oxford physics undergraduates and graduates. Registration for the next academic year is open now.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I give school talks about maths and puzzles (online and in person). If your school (or police academy) is interested please get in touch.