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Combinatorics and Linear Algebra

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Provide a linear description of all attempts you tried, including those that didn't work. Show all work that represents your process. Also, include the level of challenge you encountered. Was this a problem or an exercise and why? Provide real-life connections to each of the problems you solved. In addition, think about answering the following: Where's the math in this problem? What makes this problem challenging (if not for you, for others)? Can you provide an algorithm for solving this problem?

1. The following game is one I've played with several students of many different grades and ages. The game, as I play it with students is called "Take Two". It's known historically as Nim. The object of the game is to be the last person to take a chip. Here are the rules of the game.
~ Line up seven chips (pennies will work) in a row. O O O O O O O
~ Decide who goes first and alternate turns.
~ You may take one chip or two chips on your turn.
~ You can't pass or skip your turn.

Play the game several times with another person to determine if the game is fair. Then explain to me whether the game is fair. You must use data gathered from your experience to support your assertion. Have fun! Include in your write-up what strategies you learned as you played. Also include where the probability is in the game. Write a brief summary of the conversation and the reactions of your opponent as you played.

2. I told you in the overview for this module about my friend who buys a lottery ticket every week. I also explained that she buys extra ones when the prize is big. Choose one of the state lottery games and explain how it works. If you don't play or know about the games, you can either go on-line or talk with a friend to learn more. Then explain the likelihood of winning the game if you buy one ticket. Follow up with an explanation of how your chances are impacted by buying 20 tickets in a given week. If my friend that I mentioned in the overview played this game, would her odds be greatly improved if she purchased 19 tickets more? How would you explain it to her?

3. If you're drawing gumballs from a bag to figure out the likelihood of drawing a blue gumball, what role does putting a gumball back between times you draw have to do with the probability of an outcome? Use the example of having 6 red and 4 blue gumballs in a bag to illustrate the likelihood of drawing a blue gumball.

4. Eggbert had a basket of eggs to take to market. On the way he met a poor woman in need of food. He gave her half the eggs plus half an egg, and traveleld on. Later he met a poor man in need of food. He gave the man half of his eggs plus half an egg, and traveled on to market. There he sold the 10 eggs he had remianing. No eggs were broken at any time. How many eggs did he have to begin with?

5. Hats problem- (Sorry for the violence but I love this problem!)
In order to alleviate overcrowding in a barbaric jail, the warden decided to institute a new system of release for captives. Three men were to be lined up in a row, one behind another. Out of a bag containing 5 hats, three red and two yellow, a hat was to be drawn for each man and placed on his head. If you guessed the color of your hat correctly you'd go free. If you guessed incorrectly, you were shot. If you chose not to guess, or pass, you were sent back to an overcrowded jail. Three men are lined up. A hat is placed on each manâ??s head. The third man in line, who sees the othersâ?? hats, passes. The second man, seeing only the hat of the man in front, passes. The first man, seeing no oneâ??s hat, not even his own, knows what color his hat is, based on what the other men did. He is set free. What are the colors of the hats?

6. Two cars thirty miles apart began driving toward each other. The instant they started, a bee on the windshield of one started flying toward the other car. As soon as it reached the other car it turned and started back. The bee flew back and forth until the two cars passed each other. If each car had a constant speed of 15 mph, and the bee flew at a constant speed of 20 mph, how far did the bee fly?

7. Five logic students, Jamal, Brad, Tammy, Lee, and Minh are sitting in a row. Can you figure out the order in which they're seated from the following information?
a. Lee is the same distance from Jamal that Jamal is from Brad.
b. Minh is seated between Tammy and Jamal.
c. Brad is sitting next to Minh.
d. Minh is not seated between Brad and Tammy.

8. Sudoku- Choose a Sudoku puzzle to do, either on-line (google sudoku), through a newspaper, or from a puzzle book. Include a copy of it unsolved and solved. Explain where you used logic to solve the puzzle and what strategies you used. Be specific and give enough information that I can follow your explanation without having to read into it.

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Solution Summary

Logic, intro to probability and linear albebra problems.

See Also This Related BrainMass Solution

Linear Algebra: Find Determinant det(A)

Let a_1, a_2, ... , a_n and b_1, b_2, ... , b_n are elements in a field F. Let A be an n x n matrix with 1/ a_1 + b_1 in row i, column j. Find the det(A).

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