Write half page of each following questions:

1)Logic is shown through truth tables and used when finding whether an argument is valid or invalid. Logical reasoning can be used to solve puzzles as well. Below is a logic puzzle. Read it and respond to this discussion with your answer and a logic puzzle of your own. Be sure to post your answer after guesses have been made about the answer to your puzzle.
You live on an island where there are only two kinds of people: the ones who always tell the truth (truth tellers) and those who always lie (liars). You are accused of crime and brought before the court, where you are allowed to speak only one sentence in your defense. What do you say in each of the following situations?

If you were a liar (the court does not know that) and you were innocent. And it is an established fact that a liar committed the crime.
If you were a truth teller (the court does not know that) and you were innocent. And it is an established fact that a truth teller committed the crime.
If you were innocent and it is an established fact that the crime was not committed by a “normal” person. Normal people are that new immigrant group who sometimes lie and sometimes speak the truth. What sentence, no matter whether you were a truth teller, liar, or normal, can prove your innocence?

2) Throughout this module you have learned about numerical systems that are different from what you use day to day. Find another numerical system, research it, and share your findings with classmates. Include any information about what base is used, the symbols used, is it an additive or positional system, any special ways the users performed mathematical operations, and any other interesting facts you may discover.

3) This math problem circulated on Facebook, and may still be doing so. Facebook users were asked to post their answers in the comments, and after reviewing the posts, it was found that 74% of Facebook users got it wrong. Give your answer and discuss how you arrived at it. Explain where you think the 74% of Facebook respondents went wrong.
7 – 1 x 0 + 3 / 3 = ?

4) Word problems are created to model a real life scenario. Unfortunately, many times students cannot connect with the situations presented. Think of a time when you had to use your math skills outside of a math class. Create a word problem based on that situation. Be sure your situation is more difficult than counting change, or anything similar to that.

5) If a manufacturer can produce a certain item for $7 and sell the item for x dollars, the profit per item will be x – 7 dollars. If it is then estimated that consumers will buy 25 – x items per month, the total profit will be:
Total Profit = (number of items sold)(profit per item) = (25 – x)(x – 7)

Use this information to create a quadratic equation. Then graph your equation, and based on the graph, what value of x will create the greatest profits? What will that profit be? What part of the graph did you use to find this information? Explain how this process could be useful in other scenarios.

6) First, create a scenario that would require a permutation. Second, alter the situation in such a way that would then cause a combination to be used.

7) Read the following scenario and responses. Respond to the original question of “When would you take your turn?” and give your reasoning.
Ask Marilyn
Parade Magazine, 3 January 1999, p. 16
Marilyn vos Savant

In an earlier column (Parade, 29 November 1998, p. 26) Marilyn responded to the following question:

You’re at a party with 199 other guests when robbers break in and announce they’re going to rob one of you. They put 199 blank pieces of paper in a hat, plus one marked “you lose.” Each guest must draw a piece, and the person who draws “you lose” gets robbed. The robbers think you’re cute, so they offer you the option of drawing first, last or any time in between. When would you take your turn?

Marilyn said she would choose to draw first, explaining that “It would make no difference to my chances of losing–any turn is the same–but at least I’d get to leave this party as soon as possible.” Not all of her readers agreed, and the present column contains responses from some of them.

One letter argues for drawing first: “You said any turn is the same, but I believe that would be true only if the partygoers all had to replace the papers they drew before another selection was made. But if they keep the papers (the scenario intended by the question), wouldn’t the odds of losing increase as more blanks were drawn? If so, drawing first is best.”

Another reader argued for drawing last: “Though you have a 1-in-200 chance of getting a blank paper and not being robbed if you go first, the odds are 199 in 200 that the drawing will end with a loser (other than you) before you draw if you go last. You should go last.”

Marilyn restates her original position that it makes no difference where in the process you draw. She argues that the answer would be the same as if everyone drew simultaneously, in which case it would be more intuitive that everyone has the same 1-in-200 chance. She offers another argument based on people buying tickets for a church raffle, explaining that it makes no difference whether you buy your ticket immediately when you arrive or wait until just before the drawing.

8) Graphical representations of data are used to share information. Think of a time when you used graphical data to make a decision or share information with others. Describe this time and explain how you believe the graphical representation helped or hindered you.