Tic-Tac-Toe, Abstract Algebra and the Law Presenter: Neal Fox MATH 552 University of Virginia Instructor: Mike Hill

Levels Hypothesis Cognitive Science Cognitive Science Developed to examine problem-solving techniques Developed to examine problem-solving techniques Three levels: Three levels: –Physical –Symbolic or Computational –Semantic or Knowledge

Physical Level Anything that can be physically realizable Anything that can be physically realizable X

Physical Level Anything that can be physically realizable Anything that can be physically realizable No consideration of rules No consideration of rules XX

Physical Level Anything that can be physically realizable Anything that can be physically realizable No consideration of rules No consideration of rules No consideration of strategy No consideration of strategy XX X

Symbolic/Computational Level X Must be physically realizable Must be physically realizable

Symbolic/Computational Level XO Must be physically realizable Must be physically realizable Now INCLUDES consideration of rules Now INCLUDES consideration of rules

Symbolic/Computational Level Must be physically realizable Must be physically realizable Now INCLUDES consideration of rules Now INCLUDES consideration of rules XO X

Symbolic/Computational Level Must be physically realizable Must be physically realizable Now INCLUDES consideration of rules Now INCLUDES consideration of rules No consideration of strategy No consideration of strategy XO XO

Symbolic/Computational Level Must be physically realizable Must be physically realizable Now INCLUDES consideration of rules Now INCLUDES consideration of rules No consideration of strategy (!!!) No consideration of strategy (!!!) XO XO X

Semantic/Knowledge Level Must be physically realizable Must be physically realizable X

Semantic/Knowledge Level Must be physically realizable Must be physically realizable Consideration of rules included Consideration of rules included X O

Semantic/Knowledge Level Must be physically realizable Must be physically realizable Consideration of rules included Consideration of rules included Now INCLUDES consideration of strategy Now INCLUDES consideration of strategy XX O

Semantic/Knowledge Level Must be physically realizable Must be physically realizable Consideration of rules included Consideration of rules included Now INCLUDES consideration of strategy Now INCLUDES consideration of strategy XOX O

Semantic/Knowledge Level Must be physically realizable Must be physically realizable Consideration of rules included Consideration of rules included Now INCLUDES consideration of strategy Now INCLUDES consideration of strategy XOX OX

“Isomorphic” Games Isomorphism (A  B) Isomorphism (A  B) –Mapping must be 1-1 –Mapping must be “onto” Levels should be considered isomorphically, as well Levels should be considered isomorphically, as well

“Isomorphic” Games Bright Medium Pale

“Isomorphic” Games 1-1 (only one column/color setting mapped to for each square in traditional Tic-Tac-Toe) 1-1 (only one column/color setting mapped to for each square in traditional Tic-Tac-Toe) Onto: (each spot in traditional board maps to something in this column/color chart) Onto: (each spot in traditional board maps to something in this column/color chart)

“Isomorphic” Games Physical Level: 9 spots to symbol at start; 2 symbols can’t occupy same slot Physical Level: 9 spots to symbol at start; 2 symbols can’t occupy same slot Symbolic/Computa- tional Level: rules concerning order of placement withstand Symbolic/Computa- tional Level: rules concerning order of placement withstand Semantic/Knowledge Level: DESIRE TO WIN! Semantic/Knowledge Level: DESIRE TO WIN!

“Isomorphic” Games Physical Level: 9 spots to symbol at start; 2 symbols can’t occupy same slot Physical Level: 9 spots to symbol at start; 2 symbols can’t occupy same slot Symbolic/Computa- tional Level: rules concerning order of placement withstand Symbolic/Computa- tional Level: rules concerning order of placement withstand Semantic/Knowledge Level: DESIRE TO WIN! Semantic/Knowledge Level: DESIRE TO WIN! 1.3 of one color (row) 2.3 of one hue (column) 3.3 of different colors and adjacent/ordered hues (the two diagonals)

Legal Implications Hypothetical situations: Tic-Tac-Toe is a registered trademark. You design one of the games we discussed today. Hypothetical situations: Tic-Tac-Toe is a registered trademark. You design one of the games we discussed today. Are you in violation of copyright law? Are you in violation of copyright law?

Legal Implications Hypothetical situations: Tic-Tac-Toe is a registered trademark. You design one of the games we discussed today. Hypothetical situations: Tic-Tac-Toe is a registered trademark. You design one of the games we discussed today. Are you in violation of copyright law? Are you in violation of copyright law? Quantitative “measure of similarity” Quantitative “measure of similarity”

Patent Law & Intellectual Property Graduate degree in Mathematics, Engineering or other applicable science Graduate degree in Mathematics, Engineering or other applicable science Law School Law School VERY lucrative career opportunities VERY lucrative career opportunities On the cutting edge of scientific research On the cutting edge of scientific research Creative approaches to legal argument Creative approaches to legal argument

Further Reading Math in Tic-Tac-Toe (combinatorics, algebra, artificial intelligence, multi- dimensional games) Math in Tic-Tac-Toe (combinatorics, algebra, artificial intelligence, multi- dimensional games) Patent Law and Intellectual Property Patent Law and Intellectual Property (US Patent and Trademark Office) (American Bar Association Career Finder)

THANK YOU!