The Factor Game

1. The Factor Game 2 x 3 = 6

2. Factor Game • Two person game • You’ll need colored pens, pencils, or markers and a game sheet Objective: • Play the factor game several times and take turns making the first move. • Look for moves that give the best scores. • In your journal record the scores of each play and the strategies you find that help you to win

3. Player 1 chooses a number and circles it. • Using a different color, player 2 circles all the proper factors (not the number itself) • Player 2 circles a new number, and player 1 circles all the proper factors. • If a player picks a number that has no factors left that have not been circled, he loses the play and the circle is crossed out and the number remains in play. • The game ends when there are no numbers with uncircled factors. • The player with the greater total wins. (Add circled numbers after each play)

4. The Factor Game

5. Problem 1.1 • Students record a strategy discovered that helped to win the game. • Each student will create a table and compare it with their partner when done.

6. Playing to Win the Factor Game • Make a table of all possible first moves from 1 through 30 • For each move list the proper factors • Record your score and your opponents score

7. Problem 1.2 • Use your list to figure out the best and worst first moves. • What is the best first move? Why? • What is the worst first move? Why? • Look for other patterns in your list. Describe an interesting pattern that you find.

8. Problem 1.2 Follow-Up Questions • List all the first moves that allow your opponent to score only one point. These kinds of numbers have a special name. The are called _____________________. • Are all prime numbers good first moves? (A number is a good first move if the player picking the number scores more points than his or her opponent.) Explain your answer.

9. Problem 1.2 Follow-Up Questions • List all the first moves that allow your opponent to score more than one point. These kinds of numbers also have a special name. they are called ________________. • Are composite numbers good first moves? Explain. • Which first move would make you lose a turn? Why?

10. Answers to Problem 1.2 • 29; It gives you the largest point advantage over your opponent. • 24 and 30; Your opponent gets 12 points more than you get. • 1 is a factor of all numbers, 1 is the worst first move because you lose your play, some numbers only have 2 factors, all numbers with 2 factors are odd except 2, the numbers 4, 9, 16 and 25 have an odd number of factors, 24 is worse than 30 because you get less points with 24, and more…

11. Answers to Problem 1.2 Follow-up • 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 • Yes; all primes are good first moves because the second player gets only one point. Large primes are best. • 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28 and 30 • Some are good and some are not; it depends on the sum of the factors. 30 is a bad first move because the opponent gets 42 points. Whereas 25 is not too bad because the opponent gets only 6 points. • 1; There are no factors for the other player to choose.