Algebra: A sequence from long ago...

I keep forgetting to post our very first challenge, from way back when. Here it goes:

The Challenge: Given the first few numbers of a sequence,

1, 2, 9, 64, 625, ...

can you deduce the next number? As always, there is a pattern if you look closely enough.

Hint: Sometimes when numbers increase extremely rapidly, we say they are growing exponentially...

Algebra: A circuitous path

Imagine taking a walk in a city by a river. The river passes through the city and branches around some islands, and several bridges connect the opposite banks to the islands and each other. The image below should help you to picture this.





The challenge: Is it possible to walk about the entire city, crossing each bridge once and only once? You do not need to start and end your walk at the same place.

For the prize, you'll need either an example path or else a solid explanation of why there is no such path.

Once you have your answer, look for the pattern to explain why you can or cannot make the journey. If you can, what would happen if one of the bridges were closed? If you can't, what place(s) would a bridge be necessary in order to make it possible?

Stats: Find the shaded area

Okay, here's the old stats challenge. Imagine a regular (all equal length sides) polygon inscribed in a circle. By inscribed, we mean that all of the vertices of the polygon lie on the circle. It's not too hard to find the ratio of the polygon's area to the circle's area for some of the simpler shapes, especially a square and an octagon. I was also able to do a triangle and hexagon with some additional tricks (remember the 30-60-90 triangle?). The ratio of the polygon's area to the circle's area are as follows for these shapes:

Triangle: 3/(2*pi)
Square: 2/pi
Hexagon: 3*sqrt(3)/(2*pi)
Octagon: 2*sqrt(2)/pi

The Challenge:
For either a regular pentagon or heptagon (that's 7 sides) inscribed in a circle, find the ratio of the polygon's area to the area of the circle. For an extra bonus, find the pattern relating a general n-sided polygon's area to a circle in which it is inscribed. Note that I expect exact answers, in the forms above with square roots, pi's, and numbers in fractions, rather than simplified decimals.

This is a trigonometric problem more than a stats one, but it's important because a similar problem, the ratio of the perimeters of these polygons to the radius of the circle, formed the basis of ancient attempts to find the value of pi!

Both: The Slamming Doors

Here's this week's problem:

Imagine a long hallway, filled with closed doors. Let's say 10 doors to start. Now find 9 of your friends, and proceed as follows:

1) You walk down the hall, opening every door. Now all the doors are open

2) Your first friend walks down the hall, flipping the status of every second door. So, they will re-close the 2nd, 4th, 6th, 8th, and 10th doors. Now every other door is open or closed.

3) Your next friend walks down the hall, flipping the status of every 3rd door. So, the 3rd door will go from open (from when you opened it in step 1) to closed, the 6th door will go from closed (see step 2) to open, and door 9 will go from closed (step 1) to open.

4) Your next friend will flip every 4th door
5) Your next friend will flip every 5th door
6) Your next friend will flip every 6th door
...and so on, until the last person flips just the last door.

The Challenge: Which doors are open in a hallway of 100 doors after you and 99 friends walk through?

Hint: For this case of 10 doors, the image below should help you see what's going on in each step. Before anyone has opened or closed any doors, they're all closed (imagine the doors are painted blue, so the blue bars mean a closed door, and a white space means an open door). When you go through first and open them all, everything goes white, then person #2 flips every other door, and so on.... You can do this by hand without too much trouble, but 100 doors is way too many doors to figure out by hand -- concentrate on doing the cases up to 10, see if you can identify the pattern, and then try to figure out why that pattern shows up. The prize will only be awarded if you can explain the pattern!

Ground Rules!

Alright, now that I've finally got this thing set up, let's review the rules of the challenge:

1) Each week I will give a challenge problem in Mr. Lancaster's class. The first person to correctly give me the answer to the problem will receive a king-size Snickers bar or other candy of their choosing as a prize!

2) A guess does NOT qualify for the prize, even if the guess is correct. You must be able to explain your answer, that's the whole point!

3) The second person to come up with a correct answer will receive... half of a Snickers bar. The third person will receive half of half, or a quarter of a Snickers. The fourth person will receive an eighth, the fifth person a sixteenth, etc. (How many candy bars will all that add up to?)

4) Collaboration is encouraged, but remember: only the first person gets the whole prize, so choose wisely. Working together means you might get the answer faster, but you also have to share the wealth! On the other hand, if you work alone, a group working together might steal the whole prize before you can figure it out. Pick your poison!

5) Some weeks, Mr. Lancaster's stats and algebra classes will receive different challenges, while other weeks you will receive the same challenge. In either case, prizes are awarded independently to the two classes, so the first person from stats and the first person from algebra both get the whole Snickers. Note that the individual sections of each class are one group, so once a person in 2nd hour algebra solves the challenge, a person in 4th hour who solves it afterwards will only get half!

Good luck!
Mr. McDonald