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Arithmetic Sequences

- Math SPIRAL 2003
- University of Maryland
- Fredericka Blackwell, Spelman College
- Travis Bolden, Morehouse College
- Caiyan Chen, Trinity College
- Amina Pleasant-Bey, Trinity College

Outline of Presentation

- Describe the Arithmetic Sequence Game
- Provide strategies for finding an arithmetic

sequence in a specified game - Explain cases for which one player will win a

game - State the results of experiments
- Questions/Comments

The Game

- An arithmetic sequence is a sequence in which the

difference between any two consecutive terms is

the same. - i.e. 2, 4, 6 or -2,-5,-8
- Parameters n and k A game consists of the

counting numbers 1,2,3,,n, where k is the length

of the arithmetic sequence. (k lt n) - Players There are two players in this game,

Creator and Breaker. The Creator is the player

who is trying to make an arithmetic sequence of

length k. The Breaker is the player who is

trying to stop the creator from making the

arithmetic sequence.

Rules

- Step I Creator (Breaker) selects a number in

1,,n. - Step 2 Breaker (Creator) selects a number in

1,,n. - Step 3 If at the end of the game there is an

arithmetic sequence of length k, then the Creator

wins. Otherwise, the Breaker wins.

Demo I

- C Creator B Breaker
- k3 n5
- 1 2 3 4 5
- C
- B C
- B C C
- B C B C
- C B C B C
- Creator wins!!!

Demo II

- C Creator B Breaker
- k3 n5
- 1 2 3 4 5
- C
- B C
- B C C
- B C C B
- B C C C B
- Creator wins!!!

Strategy I k3

- Every case where n is greater than 5 can always

be reduced to the n5 case by selecting a block

of five numbers and playing accordingly. - Example
- 1 2 3 4 5 6 7 8 9 10
- or
- 1 2 3 4 5 6 7 8 9 10 11 12

Strategy II

- If the Creator can choose k-1 numbers such

that there are two unblocked arithmetic sequences

of length k containing those k-1 numbers, then

the Creator wins on his/her next move. - Lets take a look at the overhead projector!!

Strategy IIk4

- In order for the Creator to win, n must be

greater than or equal to 15. The Creator must

force all of the Breakers moves until a double

threat is created. - Lets take a look at the overhead projector!!

Wrap Around Game

- The Wrap Around Game is similar to the original

game, however, in the Wrap Around Game the

Creator can gain an arithmetic sequence by using

any order that is equally spaced. - Example
- n5 sequences can contain (5,1,2), (5,2,4),

(4,1,3) etc.

Wrap Around Demo

- C Creator B Breaker
- k3 n7
- 1 2 3 4 5

6 7 - C

- B C
- B C

C - B C

B C - C B C

B C - Creator wins the wrap around game with sequence

(5,7,2) which, unlike the original game, is not

in increasing order.

Cases

Conjectures Original Game

- the value of n required for the

Creator to win with an arithmetic sequence of

length k provided player d goes first (d is

either the Creator (c) or the Breaker (b)) . - 3( ), where k4
- 3( ), where k4

Results

- If the Creator is player II, then the value of n

for the Original case must be larger than the

value of n if the Creator is player I. - As k increases, the value of n required for the

Creator to win will increase significantly.

Acknowledgements

- Bill Gasarch, Ph.D.
- University of Maryland
- Computer Science Dept.
- Stacey Nicholls, TA
- University of Maryland
- Applied Mathematics Dept.
- Tasha Inniss, Ph.D.
- Trinity College
- Mathematics Dept.

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