Project Ideas
Braille as a Code
Rabbit Reproduction and Patterns
Coloring Maps with as Few Colors as Possible
Monty Hall's Door Problem
More Efficient Bar Crawls
The Bowl Championship Series Ranking System
Poker Probabilities
The Loan Process
Bad Statistics
Braille as a Code
The Problem: Braille is a form of print for blind individuals where raised dots on a paper represent numbers and letters. It is also a code (which is the topic we will discuss after chapter 3). This project will explore Braille as a code.
Where to get started: In the early chapters of the following book, Code : the hidden language of computer hardware and software by Charles Petzold, the Braille system of coding is discussed. There are also many internet sites dealing with Braille.
What I want to know: You should be able to give a detailed explanation of Braille and how it works as well as answer questions like, “Why is Braille a code?” and “How can blind individuals find Braille on signs in public places?.” You should also provide me with a brief history of Braille, who came up with it, when and how did it come up, etc. and perhaps even find a Braille book and try to read some yourself.
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Rabbit Reproduction and Patterns
The Problem: Here we digress into the strange world of rabbit sex. Let's start with two newborn rabbits, a male and a female. Suppose rabbits can (and do) mate at the age of one month and each month after that and produce one female and one male offspring at the end of the month they mated in. Suppose also that our rabbits are blessed with unnaturally long life (so they don’t die). How many pairs of rabbits will there be in one year?
Where to get started: Start with pencil and paper. Calculate the number of rabbits for the first 12 months and then see if you can find a pattern.
What I want to know: What is the pattern? Why is this the right pattern? What other natural occurrences of this pattern are there?
Why is this model of the rabbits not very realistic? How would you make it more realistic? Please also give a bit of history about the rabbit problem and the pattern that arises.
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Coloring Maps with as Few Colors as Possible
The Problem: Suppose we have a geographical map of the state boundaries in the US. How many different colors would you have to use to guarantee that no two states that share a border would have the same color? Turns out four colors is all you need for this map or any other map you might come up with but that was not easy to prove in general.
Where to get started: Figure out how to turn this into a graph theory problem. Practice some coloring strategies yourself. Then check out the book Four colours suffice : how the map problem was solved by Robin Wilson. See also pg 100-102 in our book and problems at the end of chapter 3.
What I want to know: What is the history of the problem? (Hint: There is a University of Illinois connection!) What about 3-colors? What are some good algorithms to assign colors to an arbitrary map?
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Monty Hall's Door Problem
The Problem: On the show Let's Make A Deal Monty Hall would show a contestant 3 doors and ask them to pick one. Behind one of the three doors was a fabulous prize and behind the other two doors was a goat. After the contestant picked a door, Monty would open one of the junk doors and ask the contestants if they wanted to stick with their choice or switch to the other unopened door. What is the best strategy?
Where to get started: Remember the Fundamental Principle of Counting from chapter 1? This will come in handy when you are figuring out the actual probabilities. Chapter 7 in the book should give you the basics about probability.
What I want to know: First have everyone in the group make their own guesses about what will happen. Then devise an experiment using dice to mimic the scenario maybe 50 times or so. What happens? Now figure out the real probability. What other real world situations use probability? Come up with some other examples where popular opinion does not match the actual probability.
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More Efficient Bar Crawls
The Problem: Bar crawls are a common weekend practice on campus. As a bar crawlee, you want to make your way around all the campus bars as efficiently as possible and you don't want to be embarrassed by going back to a bar you've already left.
Where to get started: Make a nice graph of a bunch of campus bars and the distance between them (good way to find distances is to see how far your stride is and count the number of steps you take to get from one to the other).
What I want to know: What is the most efficient path to visit all the bars once and only once? Now suppose you’re the city employee who has to go around and clean up all the litter on the streets around the bars after the bars close. What is the best route to take if the street cleaner can only clean one side of the street with each pass?
Warning: There is no need to go into any of the bars for this project so if you are not 21, don’t go in them while on assignment!
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The Bowl Championship Series Ranking System
The Problem: The BCS rankings in college football have had a rocky beginning. Many people dispute whether their rankings are fair and accurate. We want to explore a bit of the math behind the rankings.
Where to get started: I have an article entitled The Bowl Championship Series: A Mathematical Review which talks about some of the math behind the rankings. There are also MANY good sources listed in the reference list for this article.
What I want to know: How did the BCS rankings work before this year? How do they work now? Why the changes? What are the various concerns about the rankings? How does the random walking monkey rankings work? What do you think about these different algorithms?
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Poker Probabilities
The Problem: Poker is a fascinating card game that incorporates both mathematical probability and human deception. If you ever find yourself at a high stakes table, it is best to know the probabilities!
Where to get started: Make sure everyone knows how to play poker (I'm assuming each player gets dealt five cards). Play a few rounds (please keep your clothes on!) and get a feel for what hands come up the most and which come up the least. Also, remember this Fundamental Principle of Counting we talked about! Chapter 7 might come in handy here.
What I want to know: What are your predictions about the probabilities? What are the actual probabilities for each poker hand (i.e. a pair, two pairs, a flush, a full house, etc.)?
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The Loan Process
The Problem: Someday, each and every one of you will take out a loan for some purchase, be it school or a car or a house. Monthly payments can be deceptive. You may think you are getting a great deal because your monthly payments are low but over the life of the loan, you may be paying far more than you think.
Where to get started: Find out what the current rates are for several banks in town. Look through Chapter 22 of our book. The HUD (www.hud.gov) web site has lots of useful information on home loans.
What I want to know: How do loans work? If you are taking out an $150,000 loan to buy a house now, which type of loan is best and for how many years? Suppose I am restricted by the fact that I can only afford monthly payments of $650 a month. Now what is the best loan? Also, what happens if you can’t put 20% down? Can you figure out how much of the principle will you have paid off in 1 year? How about 5? (For these last two questions, you can answer them for one particular bank if you would like.)
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Bad Statistics
The Problem: Numbers are used and abused every day. People misquote and misrepresent statistics daily and often the public is duped. We should be far more sceptical of numbers than we are!
Where to get started: Peruse newspaper articles. Search the internet. When it comes to the presidential election, factcheck.org has some great nonpartisan information. Chapter 8 of our book deals with statistics.
What I want to know: A brief introduction to statistics. At least 8 different misrepresentations from different sources (please keep it as unbiased as possible). A discussion of what is wrong with these misuses of statistics and why.
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