{"id":{"repo_id":"montana-tech","oai_identifier":"oai:scholarworks.umt.edu:etd-1726"},"canonical_url":"https://search.dev.ndltd.org/etd/montana-tech/oai:scholarworks.umt.edu:etd-1726","repository":{"repo_id":"montana-tech","name":"Montana Technology","base_url":"https://scholarworks.umt.edu/do/oai/"},"display":{"title":"Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe","abstract":"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.","abstract_html":"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 ≥ 5</i> as well as on <i>PG(2, q)</i> for <i>q ≥ 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, q<sub>1</sub> ) : PG(2, q<sub>2</sub> )]</i> for all <i>q<sub>2</sub> ≥ 7</i>. Other bounds are also considered for this and other classes of nested boards.","abstract_has_math":false,"creators":["Riegel, Mary Jennifer"],"institution":"University of Montana","degree_name":"Doctor of Philosophy (PhD)","degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T03:13:45Z","subjects":["combinatorial games","positional games","projective planes","affine planes","Tic-Tac-Toe"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.umt.edu/etd/707","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Riegel, Mary Jennifer"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["University of Montana"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["combinatorial games","positional games","projective planes","affine planes","Tic-Tac-Toe"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.umt.edu/etd/707"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe"]}]}],"canonical_facts":{"dc:creator":["Riegel, Mary Jennifer"],"dc:description.abstract":["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."],"dc:identifier":["https://scholarworks.umt.edu/etd/707"],"dc:publisher":["University of Montana"],"dc:subject":["combinatorial games","positional games","projective planes","affine planes","Tic-Tac-Toe"],"dc:title":["Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe"],"dc:type":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:13:45Z"}