On form and function in board games
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1 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 / 29
2 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 / 29
3 Connection strategy games Hex is a connection" strategy game. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
4 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 / 29
5 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 / 29
6 Rules 1 Choose a colour. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
7 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 / 29
8 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 / 29
9 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 / 29
10 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 / 29
11 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 / 29
12 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 / 29
13 Game demo Hexy: a worthy opponent... Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
14 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 / 29
15 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 / 29
16 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 / 29
17 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 / 29
18 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 / 29
19 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 / 29
20 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 / 29
21 Perception of the bridge Use bridges to get further. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
22 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 / 29
23 Compare red and blue Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
24 The template Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
25 Larger templates... Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
26 Puzzle template 1 Why is red connected to the edge? Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
27 Fuzzy thinking... Look for the weakest link in the strongest chain. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
28 Background Reinvented twice, 1942 (Hein) and 1947 (Nash). Piet Hein John Nash ( ) (1928 ) Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
29 Mathematical questions Can games end in a draw? Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
30 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 / 29
31 Square Use a square board. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
32 Mathematical analysis via graph theory Connectivity graph mathematical analysis. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
33 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 / 29
34 Game of Y Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
35 Game of Y Hex is a special case of Y! Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
36 Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
37 Obscuring the maths! Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
38 My motivation for using Hex... beyond intrinsic fun Engagement activity Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
39 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 / 29
40 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 / 29
41 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 / 29
42 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 / 29
43 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 / 29
44 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 / 29
45 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 / 29
46 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 / 29
47 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 / 29
48 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 / 29
49 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 / 29
50 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 rational? 1 If yes we are done. ( 2 ) If no = = 2 = 2. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
51 Tricks and transfer Tricks questions: unique intellectual moves. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
52 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 / 29
53 Reactions to Hex Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
54 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 / 29
55 Competitive situations and play" Mathematical competitions (e.g. Olympiad) Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
56 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 / 29
57 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 / 29
58 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 / 29
59 Conclusion 1 Many games, e.g. hex, have similar mathematical structure. Chris Sangwin (University of Edinburgh) On form and function in board games December / 29
60 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 / 29
61 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 / 29
62 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 / 29
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