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From Conway's Soldier to Gosper's Glider Gun in Memory of J. H

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From Conway's Soldier to Gosper's Glider Gun in Memory of J. H From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Zhangchi Chen Université Paris-Sud June 17, 2020 Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Math in games Figure: Conway’s Life Game Figure: Peg Solitaire Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: The rule Figure: A valide move Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier Question How many pegs (soldiers) do you need to reach the height of 1, 2, 3, 4 or 5? Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier Baby case To reach the height of 1-3, one needs 2,4,8 soldiers Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier A bit harder case To reach the height of 4, one needs 20 (not 16) soldiers. One solution (not unique). Further? (Conway 1961) It is impossible to reach the height of 5. Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Peg Solitaire: Conway’s soldier 1 ' '2 '4 '3 '4 '6 '5 '4 '5 '6 '10 '9 '8 '7 '6 '5 '6 '7 '8 '9 '10 '11 '10 '9 '8 '7 '6 '7 '8 '9 '10 '11 ... '11 '10 '9 '8 '7 '8 '9 '10 '11 ... ... '11 '10 '9 '8 '9 '10 '11 ... Assign a value 'n to each position. A move towards the midline, or upwards, changes the total value by 'n(1 − ' − '2); away from the midline, or downwards, changes the total value by 'n('2 − ' − 1). Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) We wish ' > 0 (each position has positive value); n 2 ' (1 − ' − ' ) 6 0 (a valid move does not increase the value); 'n('2 − ' − 1) 0. p6 5−1 If we take ' = σ := 2 = 0:618 ::: then σ > 0; σn(1 − σ − σ2) = 0; σn(σ2 − σ − 1) < 0. In this case each valid move does not increase the total initial value. However, the total value under the line is σ5 + 3σ6 + 5σ7 + ··· = 1 (Try to verify). So finitely many soldiers cannot be enough. Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Conway’s life game Create a 2d world with lives and deaths under the rules: (brith) if a dead has exactly3 neighbours living, then it becomes alive in the next moment; (survive) if a living has2or3 neighbours living, then it survives in the next moment; (death by solitude or crowdedness) if a living has 6 1 or > 4 neighbours living, then it dies in the next moment. Figure: A ‘glider’ moving right-downwards every 4 frames Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Conway’s life game Figure: Still Life, Life Cycles of period 2 and 3 Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Other ships Figure: A ship moving rightwards every 4 frames M.I.T. 50 $ price question in 1970 Can the population of life grow without limit? Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Gosper’s glider gun The winner is Gosper’s team. Their genius ‘glider gun’ generates a glider every 30 frames. Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Making a Life Computer Build a life game in a computer, yes. Build a computer in a life game, also yes! Figure: Glider streams carry signal Figure: Like in Minecraft but earlier Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Making a Life Computer The key point is designing logical gates: NOT, AND, OR. Figure: Two gliders vanish after 4 frames. Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020) Making a Life Computer Figure: NOT gate by a glider gun Zhangchi Chen Université Paris-Sud From Conway’sAND, soldier OR gateto Gosper’s requires glider gun other In memory meeting of J. H. Conway type of(1937-2020) gliders. References Figure: Full of games Figure: John Horton Conway Wiki ‘Conway’s Soldier’ Golly, play life game on PC/Android/iPad. http://golly.sourceforge.net/ Zhangchi Chen Université Paris-Sud From Conway’s soldier to Gosper’s glider gun In memory of J. H. Conway (1937-2020).
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