Arithmetic Sequences

1
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

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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

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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.

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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.

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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!!!

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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!!!

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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

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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!!

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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!!

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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.

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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.

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Cases
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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

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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.

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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.