## Appendix B – Player Reference ### B.1 Terms Glossary - **Players:** The people playing the game. Player 1 goes first, Player 2 goes second, and then the players take alternating turns until the Observation Phase is reached. After the game is over, if a rematch is played, the players should switch who starts while keeping the same symbols, but for the purposes of this document, Player 1 is always $\triangle$ and Player 2 is always $\square$. - **Turn:** One move, as performed by one player. - **Square:** One of the nine spaces on the board, usually referred to as the top left, top, top right, left, middle, right, bottom left, bottom, and bottom right squares. - **Symbol:** Either $\triangle$ or $\square$, replacing the traditional $X$ and $O$ of classical tic-tac-toe. - **Value:** The number pointing up on a d4 die. Possible values are $1$, $2$, and $4$, and they can be coloured blank, red, or blue. - **Blank:** A number that is not coloured red or blue (in which case they are either white or gold, depending on the player). - **Piece:** A symbol that will end up on the board after the next observation phase. A piece can be either an $X$, or an $O$, or in a superposition state of both $X$ and $O$. - **State:** A quantum description of either one or two pieces. A state is defined by the d4 dice placed on the board, and always involves dice of either one colour, or one colour per player. One state is defined to be different from another if their outcomes are independent of each other. - **Classical:** A kind of quantum state a piece can be in, in which all features (square and symbol) are fully determined. (Not to be confused with an *observed* piece.) - **Superposition:** A kind of quantum state a piece can be in, in which a feature is not fully determined. The feature in question can be what square the piece is located in, or the symbol of the piece, or both. - **Entanglement:** Two pieces in superposition states that are not independent of each other are said to be entangled. - **Independency:** Two pieces that are not entangled are said to be independent. - **Plus/minus state:** An independent piece in a superposition over symbols. If the two symbols (dice) have the same sign, the state is referred to as a "plus state", and if the two symbols (dice) have opposite signs, then the state is referred to as a "minus state". - **Observation:** The process by which quantum states become classical objects, which in essense happen by "looking at" them,[^1] and in this game typically involves a roll. Observation can only happen in the observation phase. - **Observation phase:** The phase of the game that follows after the board is full, in which pieces are observed. Winners can only be declared directly following an observation phase. - **Observed:** An observed piece is drawn as a symbol on the board, at which point neither the piece, nor the square in which it sits, can be further interacted with. (Not to be confused with a *classical* piece.) ### B.2 Main Rules Following are rules common to every level of the game. 1. **One colour per turn:** For each turn, the player places either *one* or *two* d4 dice of the *same colour* on the board, where the colour is one that has not already been placed. 2. **One empty square per turn:** For each turn, at least one empty square must be spent. If the player places one d4 die, then it must be placed in an empty square with the number $4$ up. If the player places two d4 dice, then they must be placed in different squares with the number $2$ up, and either one or both squares must be empty. 3. **Max two dice per square:** At no point in the game can a square contain more than two d4 dice. 4. **No self-entanglement:** A player can not place a die in a square that already contains one of their own dice. 5. **The observation phase:** When there are no more empty squares, then there are no legal moves, and the game enters the *observation phase*. The player who placed last is then said to have *initiated* the observation phase, and is given the observation token. 6. **Deciding the game:** After observing the board, if one player has at least one three-in-a-row more than the other player, then the player with more three-in-a-rows is declared the winner. If the observation phase concludes and no winner declared, *and* there are empty squares on the board, the turn goes to the next player after the initaiting one (the player without the observation token). If the observation phase concludes with no winner declared and there are no empty squares on the board, then the game is a *tie*. ### B.3 Pieces Per Level Here is an overview of the pieces should be distributed at each level. Here, "normal d4s" refer to d4 dice where three out of four numbers are coloured white/gold, while "special d4s" refer to d4 dice where three out of the four numbers are coloured red/blue. A colour in a paranthesis refers to the majority colour of the die's numbers, not the colour of the die itself. |Level|Bank|Player 1|Player 2| |:---:|:--:|:------:|:------:| |**1:** [[TiqTaqToe with Superpositions]]|Observation Token|$5 \times 2$ normal d4s (white)|$5 \times 4$ normal d4s (gold)| |**2:** [[TiqTaqToe with Entanglement]]|Observation Token; $10 \times 2$ special d4s (red and blue)|$5 \times 2$ normal d4s (white)|$5 \times 2$ normal d4s (gold)| |**3:** [[TiqTaqToe with Interference]]|Observation Token; $10 \times 1$ special d4s (blue)|$5 \times 2$ normal d4s (white); $5 \times 1$ special d4s (red)|$5 \times 2$ normal d4s (gold); $5 \times 1$ special d4s (red)| |**4:** [[TiqTaqToe with Complementarity]]|Observation Token|$5 \times 2$ normal d4s (white); $5 \times 2$ special d4s (red and blue)|$5 \times 2$ normal d4s (gold); $5 \times 2$ special d4s (red and blue)| |**5:** [[Fully Quantum TiqTaqToe]]|" |" |"| ### B.4 Moves %% |Level|Classical Move|Superposition Move|Entangling Moves|Cancelling Moves|Spinning Moves|Skew basis|Basis change| |:---:|:------------:|:------:|:--------------:|:----:|:--------------:|:----:|:---:| |**0:** *Classical Tic-Tac-Toe*|✅|❌|❌|❌|❌|❌|❌| |**1:** [[TiqTaqToe with Superpositions]]|✅|✅|❌|❌|❌|❌|❌| |**2:** [[TiqTaqToe with Entanglement]]|✅|✅|✅|❌|❌|❌|❌| |**3:** [[TiqTaqToe with Interference]]|✅|✅|✅|✅|❌|❌|❌| |**4:** [[TiqTaqToe with Complementarity]]|✅|✅|✅|✅|✅|✅|❌| |**5:** [[Fully Quantum TiqTaqToe]]|✅|✅|✅|✅|✅|✅|✅| %% Here is an overview of the moves that are available to the players in each level of the game. |Move/Level|**0:** *Classical Tic-Tac-Toe*|**1:** [[TiqTaqToe with Superpositions]]|**2:** [[TiqTaqToe with Entanglement]]|**3:** [[TiqTaqToe with Interference]]|**4:** [[TiqTaqToe with Complementarity]]|**5:** [[Fully Quantum TiqTaqToe]]| |:---:|:------------:|:------:|:--------------:|:----:|:--------------:|:----:| |Observed Pieces|✅|✅|✅|✅|✅|❌| |Classical Move|✅|✅|✅|✅|✅|✅| |Superposition Move|❌|✅|✅|✅|✅|✅| |Entangling Moves|❌|❌|✅|✅|✅|✅| |Cancelling Moves|❌|❌|❌|✅|✅|✅| |Spinning Moves|❌|❌|❌|❌|✅|✅| |Skew-basis observation|❌|❌|❌|❌|✅|✅| |Mid-game basis change|❌|❌|❌|❌|❌|✅| We next give summaries of each move. Recall that all moves places dice of a single colour not already on the board. #### The Classical Move The player places one d4 die on the board, with a blank $4$ up, in an empty square. | |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank $4$| | |**Square**|Empty| | The classical move adds one piece to the board. #### The Superposition Move The player places two d4 dice of the same colour on the board, both with a blank $2$ up, and both in empty squares. | |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank $2$|Blank $2$| |**Square**|Empty|Empty| The superposition move adds one piece to the board. #### The Entangling Moves The player places two d4 dice of the same colour on the board, both with a blank $2$ up, in one empty square and in one square that contains one of the opponent's dice. The game then adjusts itself by halving the (blank) value of the opponent's die, and placing another die of the same colour and (halved) value in the second square. If the opponent's die originally had the value $4$ (such that it is halved to $2$), we refer to the move as *the entangling move*. If the opponent's die originally had the value $2$ (such that it is halved to $1$), we refer to the move as *the half-entangling move*. | |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank $2$|Blank $2$| |**Square**|Empty/Occupied|Occupied/Empty| The (half-)entangling move adds one piece to the board. #### The Cancelling Moves The player places two d4 dice of the same colour on the board, one with a red $2$ up and one with a blank $2$ up, in one empty square and in one square that contains one of the opponent's dice. The game then adjusts itself by halving the (blank) value of the opponent's die, and placing another die of the same colour and (halved) value in the second square. If the opponent's die originally had the value $4$ (such that it is halved to $2$), we refer to the move as *the cancelling move*. If the opponent's die originally had the value $2$ (such that it is halved to $1$), we refer to the move as *the half-cancelling move*. | |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank/red $2$|Red/blank $2$| |**Square**|Empty/Occupied|Occupied/Empty| The (half-)cancelling move does not add any pieces to the board. ##### Cancelling in later levels In Level 4 ([[TiqTaqToe with Complementarity]]) and 5 ([[Fully Quantum TiqTaqToe]]), whenever Player $O$ performs a (half-)cancelling move, the second die should be placed with a blue $2$ instead of a blank $2$ up. (Player $X$ places as before.) | $X$ (Level 4 and 5) |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank/red $2$|Red/blank $2$| |**Square**|Empty/Occupied|Occupied/Empty| | $O$ (Level 4 and 5) |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blue/red $2$|Red/blue $2$| |**Square**|Empty/Occupied|Occupied/Empty| #### The Spinning Moves The player places two d4 dice of the same colour on the board, one with a blue $2$ up and one with a blank $2$ up, in one empty square and in one square that contains one of the opponent's dice. The game then adjusts itself by halving the (blank) value of the opponent's die, and placing another die of the same colour and (halved) value in the second square. If the opponent's die originally had the value $4$ (such that it is halved to $2$), we refer to the move as *the spinning move*. If the opponent's die originally had the value $2$ (such that it is halved to $1$), we refer to the move as *the half-spinning move*. | |Die 1|Die 2| |:--:|:---:|:---:| |**Value**|Blank/red $2$|Red/blank $2$| |**Square**|Empty/Occupied|Occupied/Empty| The (half-)spinning move adds one piece to the board. ### B.5 The Observation Phase (tbd) The player points to one of the two squares and asks, "is this square mine?" and rolls. If yes, remove both dice and replace it with the player's symbol in that square, otherwise place the symbol in the other square. #### Observing in the Skew basis ### B.6 Basis Change (tbd) [^1]: If this sounds strange to you, you wouldn't be the first! For more on what it really means to "look at" a quantum state, and the consequences this has to our understanding of the world, see any book on the interpretations of quantum mechanics; I recommend [Carlo Rovelli's Helgoland](https://www.amazon.com/Helgoland-Making-Sense-Quantum-Revolution/dp/0593328892/ref=sr_1_1).