On form and function in board games Chris Sangwin School of Mathematics University of Edinburgh December 2017 Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 1 / 29
Introduction Connection strategy games: hex Board games as a model of mathematics. Games and education. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 2 / 29
Connection strategy games Hex is a connection" strategy game. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 3 / 29
Connection strategy games Hex is a connection" strategy game. Link your sides before your opponent connects his or her sides. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 3 / 29
Connection strategy games Hex is a connection" strategy game. Link your sides before your opponent connects his or her sides. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 3 / 29
Rules 1 Choose a colour. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 4 / 29
Rules 1 Choose a colour. 2 Players take turns. On each turn place one counter of your colour in a single cell within the overall playing board. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 4 / 29
Rules 1 Choose a colour. 2 Players take turns. On each turn place one counter of your colour in a single cell within the overall playing board. 3 Pieces cannot be moved. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 4 / 29
Rules 1 Choose a colour. 2 Players take turns. On each turn place one counter of your colour in a single cell within the overall playing board. 3 Pieces cannot be moved. ( swap rule later). Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 4 / 29
Rules 1 Choose a colour. 2 Players take turns. On each turn place one counter of your colour in a single cell within the overall playing board. 3 Pieces cannot be moved. ( swap rule later). 4 The four corner hexagons each belong to both adjacent sides. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 4 / 29
Classification of this game GAME: competitive scenario with rules TWO-PERSON: ZERO-SUM: players win and lose to each other FINITE: the game ends in a finite number of moves PERFECT INFORMATION: no chance, no hidden information STRATEGY: Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 5 / 29
Classification of this game GAME: competitive scenario with rules TWO-PERSON: ZERO-SUM: players win and lose to each other FINITE: the game ends in a finite number of moves PERFECT INFORMATION: no chance, no hidden information STRATEGY: Also called a connection strategy game. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 5 / 29
Game demo Hexy: a worthy opponent... Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 6 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. 3 If you can think of a strong response to your own move then look for a better one. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. 3 If you can think of a strong response to your own move then look for a better one. 4 Make connections between your pieces and simultaneously block your opponent. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. 3 If you can think of a strong response to your own move then look for a better one. 4 Make connections between your pieces and simultaneously block your opponent. 5 Play defensively: defence is also attack. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. 3 If you can think of a strong response to your own move then look for a better one. 4 Make connections between your pieces and simultaneously block your opponent. 5 Play defensively: defence is also attack. 6 Abandon areas of the board which are hopeless. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Basic strategy 1 You may place counters anywhere not just next to each other. 2 Respond to your opponent. 3 If you can think of a strong response to your own move then look for a better one. 4 Make connections between your pieces and simultaneously block your opponent. 5 Play defensively: defence is also attack. 6 Abandon areas of the board which are hopeless. 7 But never give up the game until it is clearly over! Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 7 / 29
Perception of the bridge Use bridges to get further. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 8 / 29
Bridge to an edge The same reasoning can be applied to edge templates. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 9 / 29
Compare red and blue Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 10 / 29
The 2 3 4 template Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 11 / 29
Larger templates... Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 12 / 29
Puzzle template 1 Why is red connected to the edge? Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 13 / 29
Fuzzy thinking... Look for the weakest link in the strongest chain. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 14 / 29
Background Reinvented twice, 1942 (Hein) and 1947 (Nash). Piet Hein John Nash (1905 1996) (1928 ) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 15 / 29
Mathematical questions Can games end in a draw? Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 16 / 29
Mathematical questions Can games end in a draw?... question about all games. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 16 / 29
Square Use a square board. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 17 / 29
Mathematical analysis via graph theory Connectivity graph mathematical analysis. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 18 / 29
Variations of Hex Size of board Goal (win vs lose) Shape of board: game of Y Altered connectivity Games on graphs (Shannon switching game) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 19 / 29
Game of Y Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 20 / 29
Game of Y Hex is a special case of Y! Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 20 / 29
Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 21 / 29
Obscuring the maths! Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 22 / 29
My motivation for using Hex... beyond intrinsic fun Engagement activity Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 23 / 29
My motivation for using Hex... beyond intrinsic fun Engagement activity Model for learning a skill... Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 23 / 29
My motivation for using Hex... beyond intrinsic fun Engagement activity Model for learning a skill...... and developing expertise Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 23 / 29
My motivation for using Hex... beyond intrinsic fun Engagement activity Model for learning a skill...... and developing expertise Structure within the game Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 23 / 29
My motivation for using Hex... beyond intrinsic fun Engagement activity Model for learning a skill...... and developing expertise Structure within the game Structure of the game, and similar games Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 23 / 29
Thinking ahead in mathematics Find three consecutive integers x, y and z so that xyz = x + y + z Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 24 / 29
Thinking ahead in mathematics Find three consecutive integers x, y and z so that xyz = x + y + z x = n, y = n + 1, z = n + 2 n(n + 1)(n + 2) = n + (n + 1) + (n + 2) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 24 / 29
Thinking ahead in mathematics Find three consecutive integers x, y and z so that xyz = x + y + z x = n, y = n + 1, z = n + 2 n(n + 1)(n + 2) = n + (n + 1) + (n + 2) x = n 1, y = n, z = n + 1 (n 1)n(n + 1) = (n 1) + n + (n + 1) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 24 / 29
Thinking ahead in mathematics Find three consecutive integers x, y and z so that xyz = x + y + z x = n, y = n + 1, z = n + 2 n(n + 1)(n + 2) = n + (n + 1) + (n + 2) x = n 1, y = n, z = n + 1 (n 1)n(n + 1) = (n 1) + n + (n + 1) n(n 2 1) = 3n /UoElogos/uoelo Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 24 / 29
Patterns of thought Alice looks at Bob and Bob looks at Clare. Alice is married but Clare is not. Prove that a married person looks at an unmarried person. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 25 / 29
Patterns of thought Alice looks at Bob and Bob looks at Clare. Alice is married but Clare is not. Prove that a married person looks at an unmarried person. A(M) B(?) C(U). (M) (U) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 25 / 29
Patterns of thought Alice looks at Bob and Bob looks at Clare. Alice is married but Clare is not. Prove that a married person looks at an unmarried person. A(M) B(?) C(U). (M) (U) Prove that an irrational power of an irrational number can be rational. 2 is irrational. Consider 2 2. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 25 / 29
Patterns of thought Alice looks at Bob and Bob looks at Clare. Alice is married but Clare is not. Prove that a married person looks at an unmarried person. A(M) B(?) C(U). (M) (U) Prove that an irrational power of an irrational number can be rational. 2 is irrational. Consider 2 2. 2 2 rational? 1 If yes we are done. ( 2 ) 2 2 2 If no = 2 2 2 2 = 2 = 2. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 25 / 29
Tricks and transfer Tricks questions: unique intellectual moves. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 26 / 29
Tricks and transfer Tricks questions: unique intellectual moves. But recognition and transfer is psychologically hard. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 26 / 29
Reactions to Hex Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 27 / 29
Reactions to Hex... very interesting to watch people outside their comfort zone. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 27 / 29
Competitive situations and play" Mathematical competitions (e.g. Olympiad) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 28 / 29
Competitive situations and play" Mathematical competitions (e.g. Olympiad) Societal competitions (e.g. exams) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 28 / 29
Competitive situations and play" Mathematical competitions (e.g. Olympiad) Societal competitions (e.g. exams) To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 28 / 29
Competitive situations and play" Mathematical competitions (e.g. Olympiad) Societal competitions (e.g. exams) To be trained is to be prepared against surprise. To be educated is to be prepared for surprise. Carse, Finite and Infinite Games (1986) Malcolm Swan: tests worth teaching to... Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 28 / 29
Conclusion 1 Many games, e.g. hex, have similar mathematical structure. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 29 / 29
Conclusion 1 Many games, e.g. hex, have similar mathematical structure. 2 Problematic as an engagement activity. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 29 / 29
Conclusion 1 Many games, e.g. hex, have similar mathematical structure. 2 Problematic as an engagement activity. 3 Engaging and fun, and hence popular. Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 29 / 29
Conclusion 1 Many games, e.g. hex, have similar mathematical structure. 2 Problematic as an engagement activity. 3 Engaging and fun, and hence popular. If you ve never played Hex then have a go! Chris Sangwin (University of Edinburgh) On form and function in board games December 2017 29 / 29