Unit 4 WebQuest - Internet Project
| 'Minesweeper': Secret to Age-Old
Puzzle? |
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Introduction
| Task
| Process
| Guidance
| Conclusion
| Questions
Introduction
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.''
The Task
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.
The Process
To successfully complete this project,
you will need to complete the following items.
Guidance
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?
Conclusion
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?
Questions
Lesson 11—7
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?
Lesson 12—1
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?
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