The Factor Game

Math 149

Spring 01

Paper #2: The Factor Game

In this paper you will analyze the factor game. We will play the game in class, and the rules are also listed below. You can also play the game against a computer by going to http://illuminations.nctm.org/tools/tool_detail.aspx?id=12. (Note that there is an underscore _ between the words grid and game; it’s hard to see it here where the whole thing is underlined).

The Rules of the Game: We will play the factor game in class. You can start with different sized boards: here are the rules for a board that uses the numbers 1 to 50:

Players start with a list of the integers (i.e. whole numbers) from 1 to 50. There are two players who take turns. On the first turn Player A circles a number, and then Player B circles the proper factors of Player A’s number. For example, if Player A chooses 40, then Player B would choose 1, 2, 4, 5, 8, 10, and 20. Each player gets points equal to the sum of the numbers s/he circled on the turn. In the example, Player A would get 40 points and Player B would get 1+2+4+5+8+10+20 = 50 points. Since the object of the game is to get the most points, Player A’s first move wasn’t a good one.

Now it is Player B’s turn to choose a number. Player B must choose a number that has at least one factor that isn’t yet circled (i.e. Player B cannot choose a number that will leave Player A with 0 points). For example, player B cannot choose 16 because all of its proper factors -- 1, 2, 4, and 8 -- are already circled. Player B chooses a number and then Player A circles the factors of the number that have not yet been chosen. For example, if Player B picks 44, then Player A would circle 11 and 22, but Player A would not circle 1, 2, and 4, because they had already been chosen. Each player receives points as before: Player A receives 33 points and now has a total of 73 points, and Player B receives 44 points and now has a total of 94 points.

Play continues, with the players taking turns being the one to choose the number. When it is impossible to chose a number that has at least one factor remaining, then the game is over and the player with the most points wins.

If you haven’t yet done so, play the factor game several times to start getting a feel for it. If you play against the computer, note that it is possible to beat the computer! But it is not easy.

The Paper: For this assignment you will analyze the first and second moves in the factor game. We’ll start with a small version of the game that uses the numbers 1 to 25, then move to a medium version that uses 1 to 100, and then a big version which uses 1 to 1000. You might find Excel helpful for parts of the project.

1) For the small version of the game, make a list of all the possible first moves and how many points they yield for Player A and Player B. What is the best first move for Player A? The best move for Player A is the move in which the difference between Player A’s total and Player B’s total is as large as possible, and Player A gets more points than Player B. What is the worst first move for Player A? Find two first moves that yield ties for Players A and B (you will have to go beyond 25 to do this). Can you find a third such move?

2) Now see if you can take what you learned in the small game and apply it to the medium game, without writing a table of all the numbers. What is the best first move for Player A? What is the worst first move? How could you convince a skeptic that you are right? Use what you know about factors to help you.

3) Now assume that Player A always makes the best first move. In the small game, what is the best next move for Player B? What about in the medium game? How could you convince a skeptic that you are right?

4) Now we’ll move to the large game. What is the best first move for Player A? If Player A makes this move, what is the best next move for Player B? Can you find a first move for Player A in which Player B will get at least twice as many points as Player A? You might want to try to “build” this number from small factors.

5) Extend the problem. If you played against the computer, you can write about that. What numbers did the computer like to pick? Why are they good moves? Talk about how the strategy changes near the end of the game. If you beat the computer (or came close) list the numbers you each chose. You can also extend the problem in some other way.

Notes on the Write-Up: Your write-up does not have to follow the order of the questions here. Make it so it flows from one idea to the next and can be read without this assignment sheet. Do not keep the 1), 2), etc. that appear here; write your paper as a coherent whole. Justify your reasoning and explain your problem solving techniques.

Scoring Guide

Problem Solving

Level 5 -- Does everything in Level 4 especially well. Extends the problem with insight.

Level 4 Makes an accurate table for the small game.

Finds two “tie scores.”

Finds the best first and second moves for all three games.

Finds a move where player B gets twice as many points as player A.

Includes play against the computer.

Extends the problem in a challenging way.

Level 3 Makes an accurate table for the small game.

Finds two “tie scores.”

Finds the best first and second moves for the small and medium games, and includes good thinking on the large game.

Level 2 Makes an accurate table for the small game.

Finds two “tie scores.”

Finds the best first and second moves for the small game, and includes some good thinking on the other games.

Level 1 -- Does work that is relevant to the problem, but is not up to Level 2.

Level 0 -- Work is not relevant to the problem

Reasoning

Level 5 -- Beyond level 4.

Level 4 Uses knowledge of number theory to accurately and thoroughly explain why all answers are correct.

Level 3 Uses knowledge of number theory to accurately and thoroughly explain why many answers are correct OR gives reasonably good explanations of most answers.

Level 2 -- Uses knowledge of number theory to explain why several answers are correct.

Level 1 -- Clearly shows an attempt to justify reasoning, although attempt is unsuccessful.

Level 0 -- Doesn’t explain why answers are correct or explanations contain multiple mistakes.

Communication

Level 5 -- Everything as in Level 4, but done especially well. The kind of paper you could put in a portfolio or show on a job interview as an example of your best work.

Level 4 -- The paper stands on its own. Someone who has never seen the problems before, and who is at your level of mathematics proficiency, could follow it. The problems are clearly restated in your own words or are obvious from your text, without needing to be restated.

The writing flows. It moves from one idea to another in a logical fashion. It is easy to understand. Ideas are explained fully; you tell the reader what you mean, rather than expecting the reader to guess. It’s obvious that you’ve put a lot of thought into your answer.

Your arguments are convincing. If there is a piece missing or a part you don’t understand, you tell the reader, rather than hoping s/he won’t notice.

The paper is written using complete sentences and virtually flawless grammar, punctuation, and spelling.

Written representations of the problem are appropriate and clear. Pictures and/or tables are used to make the ideas clearer.

Tables and graphs are labeled and presented in a clear fashion.

Level 3 -- The paper pretty much stands on its own. Someone who has never seen the problems before, and who is at your level of mathematics proficiency, could follow it, although s/he might have to ask one or two questions for clarification. The problems are clearly restated in your own words or are obvious from your text.

The writing flows pretty well. It moves from one idea to another mostly in a logical fashion. It might have a few rough spots, but it is generally easy to understand. Most ideas are explained clearly; you tell the reader what you mean, rather than expecting the reader to guess. The reader can tell that you’ve put a lot of thought into your answer.

Most of your arguments are convincing. If there is a piece missing or a part you don’t understand, you usually tell the reader, rather than hoping s/he won’t notice.

The paper is written using complete sentences and good grammar, punctuation, and spelling.

Written representations of the problem are appropriate and generally clear. Pictures and/or tables are used to make the ideas clearer.

Tables and graphs are labeled and presented in a clear fashion.

Level 2 -- The paper doesn’t really stand on its own. Someone who has never seen the problems before, and who is at your level of mathematics proficiency, would have to ask you some questions to be able to follow it, although if you answered these questions, s/he could follow it.

The writing has clear ideas, but doesn’t always flow well. Ideas are sometimes arranged logically, sometimes not. It’s clear that there’s some good thinking, but it isn’t always easy to understand what you mean.

Some of your arguments are convincing, some are not convincing. In places, the reader can’t tell for sure whether you understand what you’re saying or not.

There may be problems with grammar, punctuation, and spelling.

Written representations of the problem might be missing or unclear. Pictures and tables are not used or don’t always make the writing clearer.

Tables and graphs take some work to read.

Level 1 -- Paper is hard to read. It is about the problem; there is enough there to see that you did something, but not much more than that.

Level 0 -- Paper shows no understanding of the problem.

Name _______________________________

Self-Score Sheet (must be included with paper, feel free to use back side)

With whom did you work on this paper? List names.

What makes this paper your own? I.e., how did you put your own spin on the material when you wrote it up? What did you contribute towards getting a solution?

Based on the scoring guide, what score do you think you deserve for problem solving? Why? Please refer to the scoring guide in your explanation.

Based on the scoring guide, what score do you think you deserve for reasoning? Why? Please refer to the scoring guide in your explanation.

Based on the scoring guide, what score do you think you deserve for communication? Why? Please refer to the scoring guide in your explanation.

What was the most satisfying part of this assignment? What was the most frustrating part? Why?

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