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 ϕ ϕ2 ϕ4 ϕ3 ϕ4 ϕ6 ϕ5 ϕ4 ϕ5 ϕ6 ϕ10 ϕ9 ϕ8 ϕ7 ϕ6 ϕ5 ϕ6 ϕ7 ϕ8 ϕ9 ϕ10 ϕ11 ϕ10 ϕ9 ϕ8 ϕ7 ϕ6 ϕ7 ϕ8 ϕ9 ϕ10 ϕ11 ... ϕ11 ϕ10 ϕ9 ϕ8 ϕ7 ϕ8 ϕ9 ϕ10 ϕ11 ...... ϕ11 ϕ10 ϕ9 ϕ8 ϕ9 ϕ10 ϕ11 ...
Assign a value ϕn to each position. A move towards the midline, or upwards, changes the total value by ϕn(1 − ϕ − ϕ2); away from the midline, or downwards, changes the total value by ϕn(ϕ2 − ϕ − 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 ϕ > 0 (each position has positive value); n 2 ϕ (1 − ϕ − ϕ ) 6 0 (a valid move does not increase the value); ϕn(ϕ2 − ϕ − 1) 0. √6 5−1 If we take ϕ = 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)