Many Worlds Quantum Tic Tac Toe

MWQT$^3$

I've just discovered Quantum Tic Tac Toe.  This is a brilliant game designed to give players an intuition for Quantum Mechanics without requiring them to learn any complicated mathematics.

Superposition

Instead of playing in just one square each player gets to play in two squares at once.   Here player X has played in the 1st & 2nd squares on their first move, the 3rd & 6th squares on their 2nd move, and the 8th & 9th squares on their 3rd move.  Similarly player O has played on two squares each move, and the marks are subscripted to indicate when in the game play they were laid down.

Entanglement

Ultimately each pair of marks will be replaced with a single mark and the board will look like ordinary Tic Tac Toe - which is how we are able to determine a winner.  But the final location of, say, $O_2$ may need to depend on the final location of, say, $X_1$ if we are to avoid ending up with squares with multiple marks in them.  In this case the moves $X_1$ and $O_2$ behave very much as if they are entangled subsystems of the whole.

Collapse

The board can be thought of as a graph where the squares are the vertices and the moves are edges connecting two vertices.  As soon as a cycle is created the moves in its connected component  must all collapse.   For example, in the picture the move $O_6$ creates a cycle

This means that before the next move all the moves except $X_5$ must collapse.  A move collapsing is equivalent to drawing an arrow on an edge indicating which square the symbol will end up in.  But since only one arrow can point into each square, as soon as you have chosen the direction the arrow points on edge $O_6$ the arrows on all the other edges follow.  In other words there are always exactly two possible outcomes from collapsing a connected component.  In the picture above the two $O_6$'s are underlined and player X must choose in which square it should go.  The following picture shows the result of clicking on the top middle square.

Once a move has collapsed to a single square, that square cannot be played again.  If there is only one playable square remaining then playing twice in that square is permissible, and this causes an immediate collapse.

Many Worlds

Although QT$^3$ is wonderful, there's something not quite right about it.  Outside of the game we do not get to choose when a superposition of states collapses, and we certainly don't get to choose into which state it collapses.  In fact, if you subscribe to the Many Worlds Interpretation, collapse doesn't even happen.  The phenomenon known as collapse is simply entanglement, but where you are one of the entangled subsystems!

I was wondering: how could QT$^3$ be modified to be consistent with the MWI?  We need to replace the concept of collapse with a splitting of the game (including the players).  I thought about this and quickly coded it up in python.  The new game is Many Worlds Quantum Tic Tac Toe (or MWQT$^3$ for short) and it is played in exactly the same way as QT$^3$ except

1. When a cycle occurs the game splits and you are taken to the LHS game automatically
   
2. You can visit any other leaf in the game tree at any time

The tree like structure of the evolving game mirrors the evolution of the many worlds in real life and brings to mind a quote by Hugh Everett III, the founder of the MWI

"As time progresses the amoeba is constantly splitting, each time the resulting amoebas having the same memories as the parent. Our amoeba hence does not have a life line, but a life tree."

And here's the game for you to try for yourself