Below is a set of optional activities that can be used to increase your grade. The points will be added to your final total. For example, you have 24.9 points on all seven assignments, but need 25 for an A-. You really don't want to do an extensive assignment from one of the modules so you could complete one of these to get to the next grade level. They really don't have as much to do with modeling and decision making as they have to do with logical thinking and problem solving. They are meant to be fun and a diversion from the regular routine. If your like these kinds of problems and puzzles, by all means, enjoy yourself; if you don't like making your brain burn or are too busy to waste your valuable time on silly games, don't bother to even read them. If you know other problems like these, please share them with me. Judiciously incorporating some of these into your grade management plan can be a very effective way to increase you grade. Grade increases with these points are limited to one "notch", i.e., B to B+.

1. (0.15 points) Construct a graph in Excel that shows the probability of at least two people in a room of n people having the same birthday (month and day but not year). Let n go from 15 to 60. I think you will find the results interesting.
   
   
2. (0.15 points) Five men were stranded on a desert island where the only food was coconuts. These were gathered all day and by nightfall the men were so tired that they decided to wait until the next day to split up the pile of coconuts and eat. The most suspicious of the five men arose after the other four were asleep. He divided the coconuts into five equal piles, but there was one coconut left over. He took his share and hid it but gave the left over coconut to a monkey that the men had found. One by one, each of the five men arose and repeated the actions of the first man, each giving the remaining coconut to the monkey. Upon arising the next day the remaining pile of coconuts was divided into five equal shares but the monkey was disappointed, as there was none left over. How many coconuts were there in the original pile? Use spreadsheet table(s), function(s), and/or formula(s) to show your calculations.
   
   
3. (0.2 points) There is a circular field of grass with radius R. A cow is tethered to a post located on the circumference and is able to eat exactly one-half of the grass. Use a spreadsheet approach to estimate the length of the cow's tether to a precision of 5 decimal places? A calculus solution is worth an additional 0.1 points.
   
   
4. (0.15 points) You find an old parchment, describing the location of a buried pirate treasure on a desert island. The parchment contained these instructions:
   
   "On the island there are only two trees, a maple and an oak, and the remains of a gallows. Start at the gallows and count the steps required to walk in a straight line to the maple tree. At the tree, turn 90 degrees to the left and then walk forward the same number of steps. Mark the spot where you stop with a spike. Now return to the gallows and walk in a straight line, counting your steps, to the oak tree. When you reach the tree, turn 90 degrees to the right and take the same number of steps forward, placing another spike where you stop. Dig at a point exactly halfway between the two spikes and you will find the treasure."
   
   So you go the island, you find the two trees but there is no sign of a gallows. How can you find the treasure? Draw your analysis using MS Word or Excel drawing tools.
   
   
5. (0.15 points) In the Simulation module, you were given the probability of winning the pass line bet in craps as .4929293 giving the house a 1.41% advantage. Show how these numbers are calculated analytically.
   
   
6. (0.3 for r > 1.482 in., 0.2 for r > 1.481 in., 0.15 for r > 1.479 in., 0.1 for r > 1.477 in.) You have a square that is 10 inches on each side. You want to place 10 circular disks inside the square. Show an arrangement of disks that will allow for as large radii as possible.
   
   
7. (0.2) You have 12 coins which all look identical. One, however, is counterfeit and is either heavier or lighter that the other eleven. Using only a balance, (not a scale) how can you determine the counterfeit coin in just 3 balancings? For an additional 0.1 points, develop a spreadsheet that will find the "odd" coin for all 24 possibilities. List the steps in a MS Word document. (One students created an interactive Excel worksheet that automatically showed the steps for all 24 possibilities.)
   
   
8. (0.15 points) Construct and run a simulation for shuffling a deck of cards. How many perfect shuffles does it take to get a deck back to its original order? A perfect shuffle is splitting the deck exactly in half then alternating the cards from each half, always starting with the same side. (Hint: The formulas and model are quite simple once you figure out how it can be done.) For an additional 0.1 points, determine if a formula or relationship exists which can be use to predict the number of perfect shuffles required for any size deck of cards, say 20 to 100 cards?
   
   
9. (0.15) The Light Bulb Problem. A light bulb and switch factory, with 1000 employees, has an interesting quality control procedure. In the hallway on the way out of the factory there are 1000 switches to 1000 light bulbs, that are numbered 1 to 1000. All 1000 lights are off during the day. As the employees leave for day the first one that goes down the hall switches every light on. The second employee that goes down the hall moves every other switch to the opposite position (off), starting with number 2. The third employee moves every third switch to the opposite position, starting with number 3. The fourth employee moves every fourth switch to the opposite position, starting with the fourth. This procedure continues until all 1000 employees have passed by all the switches. When they are finished, how many lights are on? You should be able to find the answers with analytical logic (common sense, experimentation, pattern finding, etc.), using an Excel spreadsheet (it may get rather large), or a fairly short (about 10 lines) of programming code.
   
   
10. (0.1 to 0.4, depending on quality of analysis) This decision theory problem is deceptive in that it is much more difficult that it may first appears.
    
    Player A rolls a single, fair 6-sided die.
    Player B cannot see the die.
    Player A then tells a number between 1, 2, 3, 4, 5, or 6 to Player B. Player A may say any number, that is, he may lie.
    Player B can either roll to beat the number Player A just said (If the roll is a tie, it results in a "do over".),
    or Player B can challenge Player A meaning that if Player A was telling the truth about what he rolled, Player A wins. Otherwise player B wins.
    
    What is the optimal strategy for each player to either maximize winning or minimize losses? (Hint: Player A should lie, randomly, a certain percentage of the time. The percentage may be different for different values on the die.)