Wednesday, May 8

[Lara Hulbert]

Your Supplies:


The Question: Using one strip, what is the maximum number of regions the square paper can be divided up into? Using two strips, what is the maximum number of regions? Using three strips? Using n strips?

Answers for 1 strip and 2 strips (just to make sure we're all on the same page):

[joshuaw.bartlett]

Each strip can potentially cross all previously placed strips only one time, because lines cannot intersect more than once. When a strip crosses another, it bisects any shapes formed by that line segment. So, at most, any new strip can only add as many shapes as there are strips on the paper. So,

F(n)=F(n-1) + n where F(0)=1

F(n)=f(n-2)+(n-1)+n
=f(n-3) +(n-2)+(n-1)+n
=F(n-n)+ (n-(n-1))+ (n-(n-2)) +.... +(n-2)+(n-1)+n
= f(0)+1+2+...+n-1+n
= 1+[1+2+3+.....+(n-1)+n]

[Lara Hulbert]

Josh, I think I started somewhat similarly. Though I was working with F(n+1) = F(n) + n + 1 instead. Though I forgot that when you work with recursive formulas, you're supposed to state F(0) like that. I'm wondering if you can turn that into an explicit formula, without the ellipsis and the summation. 

[Lara Hulbert]

Here's my work: 

[Lara Hulbert]

And coworker Seth used regression analysis (which may or may not be cheating): 

[Gregory M. Hulbert]

This is an arithmetic sum problem, the nth strip can add at most n new regions. Because there is one region when there are zero strips, we have

#Regions = n(n+1)/2 + 1

[Michael]

I got the pattern during lunch and didn't have time to figure out an equation.  However, Seth's work makes me very happy lol!  

Sent from my iPhone

--
You received this message because you are subscribed to the Google Groups "Math Problem of the Day" group.
To unsubscribe from this group and stop receiving emails from it, send an email to problem-of-the-...@googlegroups.com.
For more options, visit https://groups.google.com/groups/opt_out.
 
 

[Lara Hulbert]

Wow, that's so simple. I took it way overboard.