From Conway’s soldier to Gosper’s 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:

Wiki ‘Conway’s Soldier’ , 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)