Wednesday, April 27, 2011

math problems

A couple more math problems came to mind the last couple of days. I'll let them sit a couple of days and answer in the comments.

1) With spring weather arriving I got thinking about coastal cities like Halifax and inland cities like Fredericton. Here's the problem:

In Fredericton the winter is colder; and summer is hotter than in Halifax. Is there a time during a year that the temperature is the same in Halifax and Fredericton?

2) This is based on a NYT article on math education. Here's the excerpt.
Imagine you're playing a game for money and you lost seven dollars and gained five. Don't give me a number. Just tell me: Is that a good day or a bad day?


Rob said...

The first one has a fairly elegant proof. Instinctively we know there's some crossover point where the temperatures are the same. This is a bit more formal proof.

let f(t) be the temperature in Fredericton

let h(t) be the temperature in Halifax

let d(t) = f(t) - h(t)

now there's always a temperature at any time so f(t) and h(t) are continuous functions. the difference of continuous functions d(t) will also be a continuous function

now in winter d(t) is negative as it is colder in Fredericton. in summer d(t) is positive as it is hotter in Fredericton

since d(t) is continuous there must be a point between winter and summer where d(t) is zero.

if d(t) is zero then f(t) - h(t) = 0 and then f(t) = h(t) and that proves there is a time when the temperature is the same in Halifax and Fredericton

Rob said...

Well you might have guessed question 2 was a trick question.

The problem is it encourages results oriented thinking instead of maximizing expectation and making the correct play.

For example suppose the problem is stated as follows.

Imagine you're playing a game for money with the following structure:

a fair dice is rolled twice. If the number is 1,2,3,4 you win five dollars. If the number rolled is a 5,6 you lose seven dollars. You are properly bankrolled for this game and can afford to be unlucky.

Now we have the basis for some high school level problems.

1) Is that a good day or a bad day?

2) What is the chance that you will lose money on a given day?

3) What is the chance that you will have lost money after 1 week (7 days), after 1 month (30 days), after 3 months (90 days).