Problem Solving
Math 1020Q § 005 & 011 Fall 2015


"It's not that I'm so smart, it's just that I stay with problems longer." — Albert Einstein

Question regarding the homework? Email me.

Unless otherwise noted, all submissions should be polished and written such that an adroit high school freshman would understand and be able to solve the homework in question. Please review the writing checklist from your book (pg. 96) when considering your final write up.

Day 1 (8/31):
Automathography: Compose a short paper (no longer than 2 pages) on your personal history with mathematics. Topics can include: your opinions on the subject; classes you've loved or hated based around math; math teachers you've had who were exceptional, or dismal; and so on. I ask that you please provide explanations of your opinions for I'm very interested in why you think these things. Including a few things about yourself to help me get to know you better is happily welcome and encouraged!

      Due: Day 2 (9/2).

Day 3 (9/4):
Heap Letter: Write a letter to a family member or friend explaining the game Heap of Beans (the game we played on the first day of class). After explaining the mechanics of the game, explain a winning strategy. I'm less worried about the correct winning strategy (as we all know it) and am more concerned with a clear and coherent explanation of the game and the winning strategy; especially why the strategy works. I will be assuming the reader of your letter has no mathematical background, or bean-game background. I hope that after reading your letter, they will both understand how to win any game of Heap of Beans that comes their way as well as why they can win any game.

      Due: Day 4 (9/9).

Day 5 (9/11):
Symmetry: Determine the number of symmetries of an equilateral triangle, a square and conjecture the number of symmetries of a regular n sided polygon. Write-up your solutions, neatly, detailing how you determined each number. Illustrations and justifications for each symmetry are encouraged.

      Due: Day 7 (9/16).

Day 7 (9/16):
Prom Problem: The problem statement can be found here.

      Due: Day 14 (10/2).

Day 16 (10/7):
Fifty Pirates: Fifty pirates, all of different ages, find a treasure trove of booty buried on Rapa Nui. They manage to plunder hundereds and hundereds of identical gold coins. The only trouble now is to divide them up. The pirates settle upon the following procedure. The youngest divides the coins among the pirates however they wish, then all 50 pirates vote on whether they are satisfied with the division. If at least half vote yes, the division is accepted. If a majority votes no, the youngest pirate is forced to walk the plank and the next youngest gets to try to divide the loot among the remaining 49 pirates (including herself). Again they all vote, with the same penalty if the majority votes {\bf no}. And so on. Each of the pirates is logical and very greedy. They will always act in their own self-interest, ignoring the interest of the group, fairness, etc. Given all this, how should the youngest of the 50 pirates divide the loot?

Please submit a written solution detailing your thought processes and justifying your work in accordance with the default homework standards mentioned above.

      Due: Day 19 (10/14).

Day __:
Should You Switch?: You are a contestant on a game show where one million dollars is hidden behind one of three closed doors. There is a goat behind either of the other two doors. You guess door 1, but before opening that door the game show host opens door 2 and shows you that a goat was hidden there. He offers you the chance to switch your guess to door 3. Should you switch?

Write up a detailed explanation as to whether you think you should switch or not. Report on and use the knowledge gained from the experiment we ran in class.
If you were absent you may want to run this experiment with someone to gain experimental data or contact someone who has attended recently and discuss the experiment conducted in class.
You may justify your conclusion using either probabilities or odds. Make sure you understand the difference and justify why you consider the odds or probabilities to be what you say they are.

Next, consider the same game involving 10 doors (9 goats, 1 bag of a million dollars), but after you make your initial choice, the host opens 8 doors. Should you switch in this case? Does this game differ? Why or why not? Justify your conclusions similar to above. Provide detailed information about how you arrived at any conclusion you made.

Please refer to the writing checklist when considering your final write up. Submissions should be in short essay format (paragraphs, complete sentences, etc).

      Due: Day 23/24. Section 005 Due 10/23. Section 011 Due 10/26.

Day __: (11/9):
Grappling with Groups: The problem statement can be found here.

      Due: 11/18.

Day __: (12/2):
Extra Credit. Blue Eyes: A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.

The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:
``I can see someone who has blue eyes.''

Who leaves the island, and on what night?

There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question, and the answer is logical. It doesn't depend on tricky wording or anyone lying or guessing, and it doesn't involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she's simply saying ``I count at least one blue-eyed person on this island who isn't me.''

And lastly, the answer is not ``no one leaves.''

      Due: 12/9.

"Difficulties strengthen the mind, as labor does the body." — Lucius Annaeus Seneca

"Understanding is joyous." — Carl Sagan