*Maureen T. Carroll and Steven T. Dougherty*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0013
- Subject:
- Mathematics, History of Mathematics

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the ...
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This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.Less

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.

*João P. Hespanha*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175218
- eISBN:
- 9781400885442
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175218.003.0017
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum discrete time dynamic game in a state-feedback policy. It begins by considering solution methods for two-player ...
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This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum discrete time dynamic game in a state-feedback policy. It begins by considering solution methods for two-player zero sum dynamic games in discrete time, assuming a finite horizon stage-additive cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. The discussion then turns to discrete time dynamic programming, the use of MATLAB to solve zero-sum games with finite state spaces and finite action spaces, and discrete time linear quadratic dynamic games. The chapter concludes with a practice exercise that requires computing the cost-to-go for each state of the tic-tac-toe game, and the corresponding solution.Less

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum discrete time dynamic game in a state-feedback policy. It begins by considering solution methods for two-player zero sum dynamic games in discrete time, assuming a finite horizon stage-additive cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. The discussion then turns to discrete time dynamic programming, the use of MATLAB to solve zero-sum games with finite state spaces and finite action spaces, and discrete time linear quadratic dynamic games. The chapter concludes with a practice exercise that requires computing the cost-to-go for each state of the tic-tac-toe game, and the corresponding solution.