## Tic Tac Toe

Tic Tac Toe

[[[ Nik’s version: (with alternation built in) ]]]

This version is incorrect:

TicTacFunction[k_Integer]:=Function[list,With[{a=First[Mod[Commonest[DeleteCases[Flatten@list,0]],k]+1,1]},ReplacePart[list,#a]&/@Position[list,0]]]

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TicTacFunction[k_Integer,t_Integer]:=Function[list,With[{a=Mod[t,k,1]},ReplacePart[list,#a]&/@Position[list,0]]]

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SW’s version:

CheckWin[list_]:=Catch[Module[{n=First[Dimensions[list]],u},Do[u=Part[list,i,All];If[!AllTrue[u,#0&]&&SameQ@@u,Throw[First[u]]],{i,n}];Do[u=Part[list,All,i];If[!AllTrue[u,#0&]&&SameQ@@u,Throw[First[u]]],{i,n}];u=Table[list[[i,i]],{i,n}];If[!AllTrue[u,#0&]&&SameQ@@u,Throw[First[u]]];u=Table[list[[-i,i]],{i,n}];If[!AllTrue[u,#0&]&&SameQ@@u,Throw[First[u]]];False]]

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Clear[CheckWin]

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Nik’s version:

CheckWin[board_]:=With[{dims=Dimensions@board},First[Keys@Select[KeySelect[Merge[Map[Counts,Join[board,Transpose@board,{Diagonal@board,ResourceFunction["Antidiagonal"]@board}]],AnyTrue[EqualTo[First@dims]]],GreaterThan[0]],Identity],False]/;Length[dims]===2&&SameQ@@dims]

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PlotBoard[list_String,sz_:40]:=PlotBoard[ToExpression[list],sz]

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PlotBoard[list:{{_Integer..}..},sz_:40]:=ArrayPlot[list,ImageSizesz,ColorRules{0White,1Orange,2Purple},MeshTrue]

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PlotBoard[i_Integer,sz_:40]:=Framed[Framed[PlotBoard[{{i}},sz]]]

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TicTacWinAssociation[k_:2]:=<|"StateEvolutionFunction"(With[{t=First[#],s=Last[#]},Thread[t+1->If[IntegerQ[s],{s},With[{w=CheckWin[s]},If[w=!=False,{w},TicTacFunction[k,t+1][s]]]]]]&),"StateEquivalenceFunction"SameQ,"StateEventFunction"Identity,"EventDecompositionFunction"Identity,"EventApplicationFunction"Identity,"SystemType""None","EventSelectionFunction"Identity|>;

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TicTacWinBeforeAssociation[k_:2]:=<|"StateEvolutionFunction"(With[{t=First[#],s=Last[#]},Thread[t+1->If[IntegerQ[s],{s},With[{w=CheckWin[s]},If[w=!=False,{w},With[{c=CheckWin[#]},If[c=!=False,c,#]]&/@TicTacFunction[k,t+1][s]]]]]]&),"StateEquivalenceFunction"SameQ,"StateEventFunction"Identity,"EventDecompositionFunction"Identity,"EventApplicationFunction"Identity,"SystemType""None","EventSelectionFunction"Identity|>;

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TicTacAssociation[k_:2]:=<|"StateEvolutionFunction"(With[{t=First[#],s=Last[#]},Thread[t+1TicTacFunction[k,t+1][s]]]&),"StateEquivalenceFunction"SameQ,"StateEventFunction"Identity,"EventDecompositionFunction"Identity,"EventApplicationFunction"Identity,"SystemType""None","EventSelectionFunction"Identity|>;

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ResourceFunction["MultiwaySystem"][TicTacAssociation[2],{Table[0,2,2]},1,"StatesGraph","StateRenderingFunction"(Inset[PlotBoard[Last[#2],30],#1]&),"IncludeStepNumber"True,GraphLayout"SpringElectricalEmbedding"]

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ResourceFunction["MultiwaySystem"][TicTacAssociation[2],{Table[0,2,2]},2,"StatesGraph","StateRenderingFunction"(Inset[PlotBoard[Last[#2],30],#1]&),"IncludeStepNumber"True,GraphLayout"SpringElectricalEmbedding"]

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Out[]=

ResourceFunction["MultiwaySystem"][TicTacAssociation[2],{Table[0,2,2]},3,"StatesGraph","StateRenderingFunction"(Inset[PlotBoard[Last[#2],30],#1]&),"IncludeStepNumber"True,GraphLayout"SpringElectricalEmbedding"]

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Out[]=

ResourceFunction["MultiwaySystem"][TicTacAssociation[2],{Table[0,2,2]},4,"StatesGraph","StateRenderingFunction"(Inset[PlotBoard[Last[#2],30],#1]&),"IncludeStepNumber"True,GraphLayout"SpringElectricalEmbedding"]

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One unit in branchial space = one move was made differently

All specific games are paths (which are geodesic paths) in this graph....

https://en.wikipedia.org/wiki/Quantum_tic-tac-toe

In ordinary QTTT, one is doing operations (including measurement) separately on every square.

We can color this picture by orange and purple wins, and ties.

A failure to entangle in the branchial graph (AKA an event horizon) is a boring game whose outcome is very predetermined...

NOTE: 2 in a row diagonally is not considered a win here

The win states are generational states because “something has happened” at every element.

There are no branchlike separated configurations because every square is spatially distinct.

#### A “complete completion” would give only one possible outcome for the game

A “complete completion” would give only one possible outcome for the game

Each outcome in the “real game” is equivalence class of states.

### Definition of a quantum experiment: divide the states of the universe into equivalence classes labeled by the results of the measurement

Definition of a quantum experiment: divide the states of the universe into equivalence classes labeled by the results of the measurement

#### [ Somewhat weird that in the game there is not time invariance, because of back-and-forth moves ]

[ Somewhat weird that in the game there is not time invariance, because of back-and-forth moves ]

[Could avoid by simultaneous doing O and X moves]

## Double-move case

Double-move case

Can have a double win in this case....

## Non-physicality of reference frames ↔ general covariance

Non-physicality of reference frames ↔ general covariance

There is no observational difference between a different coordinatization and a different spacetime geometry / gravitational field

Geometry of branchial/multispace determines the particular “coordinatized measurement”

#### E.g. Wheeler-de Witt should be able to be written as a tensor eqn

E.g. Wheeler-de Witt should be able to be written as a tensor eqn

#### Path integral “as a tensor equation”

Path integral “as a tensor equation”

The standard Lagrangian action principle is an extremization of an action

R Sqrt[g] relativistic case

(R + T) Sqrt[g]

R Sqrt[g] relativistic case

(R + T) Sqrt[g]

Einstein equations relate R and T; therefore for extremization you can extremize

In QM case, there is a R somewhere in the Fubini-Study of Hilbert space .... but you can apply the equation of motion to get rid of it, and

#### Add in completions ... get Born rule

Add in completions ... get Born rule

#### Will the quantum and classical cases behave the same? [What about biased case?]

Will the quantum and classical cases behave the same? [What about biased case?]

#### Score difference ~ phase difference

Score difference ~ phase difference

### Analog of double slit: where did the win happen on the board?

Analog of double slit: where did the win happen on the board?

Given a board position, if I am going to place another mark, how are my possible moves “rated” in score/phase...