Year of Award


Document Type


Degree Type

Doctor of Philosophy (PhD)

Degree Name


Department or School/College

Department of Mathematical Sciences

Committee Chair

Jennifer M. McNulty

Commitee Members

Mark Kayll, George McRae, Nikolaus Vonessen, Michael Rosulek


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


University of Montana


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 AG(2, q) and PG(2, q) 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 AG(2, q) for q ≥ 5 as well as on PG(2, q) for q ≥ 3. 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 [AG(2, q1 ) : PG(2, q2 )] for all q2 ≥ 7. Other bounds are also considered for this and other classes of nested boards.



© Copyright 2012 Mary Jennifer Riegel