Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe

In this dissertation we explore variations on Tic-Tac-Toe. We consider positional games played using a new type of move called a hop. A hop involves two parts: move and replace. In a hop the positions occupied by both players will change: one will move a piece to a new position and one will gain a piece in play. We play hop-positional games on the traditional Tic-Tac-Toe board, on the finite planes <i>AG(2, q)</i> and <i>PG(2, q)</i> as well as on a new class of boards which we call nested boards. A nested board is created by replacing the points of one board with copies of a second board. We also consider the traditional positional game played on nested boards where players alternately occupy open positions. We prove that the second player has a drawing strategy playing the hop-positional game on <i>AG(2, q)</i> for <i>q &#x2265 5</i> as well as on <i>PG(2, q)</i> for <i>q &#x2265 3</i>. Moreover we provide an explicit strategy for the second player involving weight functions. For four classes of nested boards we provide a strategy and thresholds for the second player to force a draw playing a traditional positional game as well as the new hop-positional game. For example we show that the second player has a drawing strategy playing on the nested board <i>[AG(2, q₁ ) : PG(2, q₂ )]</i> for all <i>q₂ &#x2265 7</i>. Other bounds are also considered for this and other classes of nested boards.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Grantor dc:publisher

University of Montana

Year

2012

Author dc:creator

• Riegel, Mary Jennifer

• combinatorial games
• positional games
• projective planes
• affine planes
• Tic-Tac-Toe

Repository record dc:identifier

https://scholarworks.umt.edu/etd/707

OAI identifier oai:identifier

oai:scholarworks.umt.edu:etd-1726