Ping Pong Balls in a Room Here s the question we re going to ponder: Approximately how many ping pong balls can fill our classroom? 1. An approximation strategy: Figure out how many balls will fit on the floor of the room, and then imagine filling the room up in layers. To actually put this idea into practice, answer these questions with your group. a. How do you visualize a layer of ping pong balls covering the floor? Discuss some different patterns that are possible. Draw some diagrams of the different possibilities here. b. Of the patterns you considered, which do you expect to give the most accurate estimate? Which do you expect to be the easiest to work with? Discuss the tradeoffs, and then choose one pattern as a group. c. Estimate how many ping pong balls would fit on the floor of the classroom in the pattern you chose. Use whatever measuring tools you have handy. If no one has a ruler, for example, you can use other objects of known size: 8.5 x 11 inch paper; floor or ceiling tiles; 3 x 5 index cards; body parts, etc. Describe any measurements or estimates you made and explain how you came up with your total estimate for pingpong balls on the floor, and what that estimate is. d. Do you expect your estimate to be high? Low? Or can you not tell? Discuss as a group and record the answer below. 1
e. Now that you know how many pingpong balls fit in one layer, come up with a way to estimate the number of layers that will fit in the room. Use that to estimate the number of pingpong balls that will fit in the room. Explain below what your method was and what answer you found. Also explain whether you think the estimate is high, low, or are not sure. 2. A different strategy: If you know the volume of the room and the volume of a ping pong ball in some units (say quarts, for example) you could divide one by the other. a. Discuss the pros and cons of this strategy. How could you estimate the volume of the room? What units of measurement would be convenient? Answer the same questions for a ping pong ball. If you could measure those accurately and divide on a calculator, do you expect the result to be an accurate estimate of how many ping pong balls would fit in the room? Would your answer be too high or too low, and why? 2
b. Actually carry out this plan: estimate the volume of the room and the volume of a ping pong ball (or perhaps a cube that would just contain a pingpong ball), and then divide to estimate the number that would fit into the room. c. Compare your answers from the two approaches you have followed above. Discuss whether you can have any confidence in the answer, or how far off the answer might be. Compare your confidence in these answers with the confidence you felt at the beginning for the guesses that the class came up with. 3
Identical hair-follicle twins?! Here s the next question we re going to ponder: Are there two (non-bald) people in the world who have exactly the same number hairs on their body? On the surface this appears to be almost impossible to answer. But there is an approach using the same kind of estimation we have already used that can provide quite a convincing answer. Before jumping in, though, let s use an important problem solving strategy: Pose and try to solve a similar problem that is smaller and/or easier. Here are some suggestions that we can begin with before we tackle the larger problem. As you re answering these questions, think about what larger ideas could be applied towards solving the identical hair-follicle twins. 1. Without asking everyone, do you think that two people in this class have the same birthday? Do you know that two people in this class have the same birthday? Explain your reasoning. 2. Do you think or know that two students at American University have the same birthday? Some of you may say Yes because you happen to know two people at the school with the same birthday. If that s the case, do you think or know that two students at the American University in Bulgaria (yes, that s a real school with 961 students) have the same birthday? Explain your reasoning. 3. What s the fundamental principle that allowed you to answer questions 1 and 2? How can it help you to answer the identical hair follicle twin question? What would you have to do next? 4
Identical hair-follicle twins continued 1. There are currently about 6.6 billion people on earth. Suppose we could estimate the number of body hairs on any individual. How could this gift and knowledge of the earth s population answer whether we know if there are two people who are identical hair-follicle twins? Would we want to consider a very hairy person, a not-so-hairy person, or an average person to help us answer this question? Explain your reasoning. 2. Now comes the hard part how would you estimate the number of body hairs on a person. Be explicit! Describe what kinds of information you would need to get started. What assumptions will you make in order to get a handle on this problem? How would you proceed to get an actual number in the end? 5
Sampling of a square inch hair follicles in a beard Here is a close-up photo of a square inch of skin from a typical human being. Estimate how many hair follicles there are in this region. The number doesn t have to be exact, but rather around the right amount. How can this help us to figure out the maximal number of hairs a human could have? 6
Additional questions about the Pigeonhole Principle 1. The pigeonhole principle states that if there are more items than pigeonholes, and if every item is in a pigeon hole, than at least one pigeonhole must contain at least two items. Identify in each of the two previous problems (identical hair-follicle twins and two people at American university having the same birthday) what were the items and what were the pigeonholes. Explain your reasoning. 2. If there are more items than pigeonholes, do we know that each pigeonhole contains at least one item? Is this the same statement as the pigeonhole principle? Why or why not? 3. If there are 350 people in an auditorium, it is likely that two of them will share a birthday? Do we know that two of them will share a birthday? Is the distinction ever important? 7
4. How many people in an auditorium would you need in order to guarantee that at least three people share a birthday? What about four people? Explain how you figured this out. Important: Don t be afraid to use logic and think!!! The pigeonhole principle doesn t explicitly cover this. How can you modify it? (Good strategy: work with a smaller problem than birthdays. For example, how many people do you need to have in an auditorium to know for sure that at least 2 have the same final social security number digit? At 3 people? Then apply what you learned to the birthday question.) 5. The question How many hairs are on a human body? is an example of a Fermi question (or Fermi problem), named in honor of the physicist Enrico Fermi. If you are curious about this topic, do a google search on Fermi question or Fermi problem and see what you can find. 8