Unit 4 WebQuest - Internet Project
|'Minesweeper': Secret to Age-Old Puzzle?
Introduction | Task | Process | Guidance | Conclusion | Questions
USA Today, November 3, 2000
Minesweeper, a seemingly simple game included on most personal computers, could help mathematicians crack one of the field's most intriguing problems.
While some people may pass hours mindlessly playing Minesweeper, mathematicians are toiling over a larger version of the game, hoping to solve a problem so confounding that an institute offered $1 million to anyone who could crack it.
The buzz began after Richard Kaye, a mathematics professor at the University of Birmingham in England, started playing Minesweeper.
"I'm always interested in games with math elements. Math and games go together brilliantly,'' Kaye said. "I realized there was probably some nice mathematics behind it. But I didn't know what I was looking for.''
Minesweeper, which is included with the Windows operating system, is a game in which players try to figure out which squares of a grid contain computerized mines. A number in each square indicates how many mines are in the squares around it.
After playing the game steadily for a few weeks, Kaye realized that Minesweeper, if played on a much larger grid, has the same mathematical characteristics as other problems deemed insolvable.
In fact, he said, Minesweeper could help in the quest for one of mathematics' white whales: the solution to what is known as the "P vs. NP'' problem. The problem has been around for about 30 years.
The problem attempts to determine whether questions that seem to be unsolvable within a reasonable period of time might actually have a relatively simple way of being solved, possibly by computer.
Kaye said that if someone were to figure out an algorithm for determining all combinations of mine placement in a large-scale version of Minesweeper, that person will have solved the P vs. NP problem.
The Clay Mathematics Institute in Cambridge, Mass., has offered $1 million to anyone who can solve it.
The discovery could have a wider impact.
"If there was a way of playing Minesweeper efficiently, then there would also be a way of cracking codes efficiently,'' Kaye said.
Ian Stewart, a research mathematician who teaches at the University of Warwick in England, said: "It's surprising that such a simple game would put us at such a frontier of mathematics. But the big questions in math are not very far below the surface of everyday life.''
Arthur Jaffe, president of the Clay Mathematics Institute, said he was already a fan of Minesweeper before he heard about Kaye's research. He plays the game on nights when he has trouble falling asleep.
"I told my 14-year-old daughter about it,'' he said. "She was just amazed it was educational.''
You are a college professor at a large university. One requirement of your job is to make presentations regularly at educational conferences in the U.S. or other countries. You have been selected to make a presentation at a mathematics conference whose theme is past mathematicians and mathematics history. You need to prepare a one-hour talk about a mathematician of the past who contributed to the field of discrete mathematics. You will place the materials for your presentation into a binder to use as reference during your presentation. Be sure that your binder contains the following:
- an outline for your talk, including the name of the mathematician, the time period the person lived, and a history of the person's life and contributions to the area of discrete mathematics;
- at least three sheets that can be made into transparencies to enhance your presentation;
- at least one example of a mathematical idea or problem that the person worked on in discrete mathematics. This example will be one of your transparencies.
You will get some ideas about mathematical ideas or problems that could be used in a talk from the exercises in Unit 4 in your textbook.
To successfully complete this project, you will need to complete the following items.
Here are some additional questions and ideas you may want to consider for your project.
- Prepare a timeline for the mathematics topic you choose. For example, you might plot the dates for some of the major advancements in probability.
- Determine what other mathematicians worked closely with the mathematician that you are highlighting. On what projects did they collaborate?
- When and how were special mathematics symbols developed, for example, summation notation?
Here are some ideas for concluding your project.
- Present your project to your class or at a family night. Consider preparing your presentation with presentation software that includes video and audio.
- Present the information on a Web page. Have other students critique your project and help you to make improvements to your project.
- Write a research paper on the mathematician you selected. Include at least one mathematics idea or problem the person worked on.
- Interview a mathematics professor at a university. What training is needed to pursue this career? What expectations are there for maintaining a position as a professor?
Tahani is preparing a presentation on Blaise Pascal (1623-1662), who is credited with the discovery of the famous pattern of numbers known today as Pascal's triangle. She plans to show Pascal's triangle on a transparency and highlight the various applications of this triangle to discrete mathematics. In her research she finds this interesting application that is not credited to Pascal.
Neatly construct an equilateral triangle, like the one below, in which you can display the first eight rows of Pascal's triangle. Finish filling in the rows.
- After you have completed Pascal's triangle, shade in all the numbers that are not odd and all of the triangles with no numbers. Describe the pattern. Can you identify this figure? (Hint: See Lesson 11-6B.)
- What is the ratio of the shaded area to the non-shaded area of the triangle in part b?
Patricia is preparing a presentation on Pierre de Fermat (1601-1665), who was a practicing lawyer, but studied mathematics as a hobby. He corresponded with Pascal over one of the first important probability problems, often referred to as the "problem of the points." According to Howard Eves, author of a mathematics history book, the problem can be stated as follows.
Determine the division of the stakes of an interrupted game of chance between two supposedly equally-skilled players, knowing the scores of the players at the time of interruption and number of points needed to win the game. (Eves, p. 288)
Fermat provided a solution for one case of this problem. He assumed that player A needed 2 points to win and player B needed three points. (The winner scores 1 point for each win.) You can see that in at most 4 more plays one of the players must win.
- Let a represent a win by player A and b represent a win by player B. Finish this list of possible outcomes for the next 4 plays.
a a a a
b a a a
a b a a
- How many possible outcomes are there?
- For how many outcomes does player A score at least 2 points? For how many outcomes does player B score at least 3 points?
- In what ratio should the remaining money be divided to be fair according to the possible outcomes?