Tanya Khovanova's Math Blog

Fire, Help!

11th February 2009, 01:39 pm

I do not remember where I dug this logic puzzle out from:

Folks living in Trueton always tell the truth. Those who live in Lieberg, always lie. People living in Alterborough alternate strictly between truth and lie. One night 911 got a call: “Fire, help!” The operator couldn’t identify the phone number, so he asked, “Where are you calling from?” The reply was Lieberg.
Assuming no one had overnight guests from another town, where should the firemen go?

After you have solved this problem, you will see that the sentence “Fire, help!” is true. I wonder, if this statement were a lie, how would we interpret it? It could be that it is just a joke and there is no fire and therefore no need for the police. Or it could be that help is needed — perhaps for a robbery, but not for a fire.

However, when you solve this puzzle, you’ll find out that the “Fire, help!” statement is true, so you do not need to wonder what it would mean if the statement were a lie. But I wondered, and as a result I invented several new puzzles.

Here is the first one:

Let’s say that people will call the police only if something is happening and there are only two things that could be happening: fire or robbery. Suppose that when people calling the police need to lie, they replace fire with robbery, and vice versa. Suppose also that when asked about location, people will not say, &quotI am not from Lieberg,” as they could have, but will always reply with one word, which is a name of one of three towns. So, there are two possibilities for the first statement — Fire, Help! or Robbery, Help! — and three possible answers to the question about location. — Trueton, Lieberg and Alterborough. My puzzle in this case is: What answer to the location question will give the biggest headache to the police?

We can branch the original puzzle out in a different way. Here is my second puzzle:

We can assume that only fire, not robbery, could happen in this remote place and when people call the police they either say “Fire, help!” or “We do not have a fire, thank you for your time.” Let us again assume that people will call the police only if something is happening. In this case, what combinations of first and second sentences of the callers will never happen?

My third puzzle is a more complicated version of the second puzzle:

Let’s assume that fires happen with the same probability in every town and that the cost of sending a team of firefighters is identical for every town. Furthermore, the ferocity of all the fires is minuscule and the cost of sending a team is the same, whether or not it turns out that there is a fire. If the police think that the call could have come from several places, they send several teams and the cost multiplies accordingly. The police are obsessed with making charts and the most important number they analyze is the cost they incur, depending on the content of the first sentence of each call.
Given that there are frequent fires, what could the ratio of the cost to police for the calls “Fire, help!” be compared to those that begin with “We do not have a fire, thank you for your time”?