## Appendix C – Player Reference **NB: WORK IN PROGRESS** ### 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 their symbols, but for the purposes of this document Player 1 is always $\triangle$ and Player 2 is always $\square$. - **Bank:** The collection of dice used by the board to *adjust* itself. - **Turn:** One move 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, and sometimes numbered 1 to 9. - **Symbol:** Either $\triangle$ or $\square$, replacing the $X$ and $O$ symbols of traditional tic-tac-toe. - **Value:** The number showing on a die. Possible values are $4$, $2$, $1$, $-2$, and $-1$. - **Piece:** A symbol that will end up somewhere on the board after the next observation phase, whose quantum state is represented during play by one or more dice. There is at any point in time an exact number of pieces on the board; only their symbol and placement on the board may be undetermined before observation. All moves add one piece to the board, except the cancelling and half-cancelling moves, which add no pieces to the board. - **State:** A quantum description of either one (unentangled) or two (entangled) pieces. Quantum states are uniquely specified by the dice on the board. One state is said to be independent of another (unentangled) if their outcomes are independent of each other. - **Classical:** A kind of quantum state a piece can be in, in which position and symbol are both fully determined (under a standard observation). Not to be confused with an *observed* piece, a classical piece can still be interacted with. - **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. Two pieces can be either fully entangled or *half*-entangled. - **Cancelling:** A square is said to be *cancelled* if it contains two quantum probabilities that have been subtracted in such a way that the result is $0$, in which case it is guaranteed to end up empty upon observation. A square can also be *half-cancelled*, in which case it will be observed to be empty with $97\%$ probability. - **Interacting:** Any move that involves placing a die in a square already occupied by the opponent is said to be *interacting*. The (half-)entangling and (half-)cancelling moves are interacting. - **Adjustment:** After interacting moves, the board *adjusts itself*, halving certain dice values and moving dice from the bank to the board. - **Symbol superposition:** A piece which is in a determined square but in a superposition over symbols. If the values corresponding to each symbol have the same sign (e.g. $\triangle + \square$), the state is referred to as a *plus* ($+$) state, and if the two values have opposite signs (e.g. $\triangle - \square$), 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", or *observing*, them.[^1] This typically involves a roll of the d6 observation dice. Observation only occur as part of the observation phase. - **Observation phase:** The phase of the game that follows once the board is full, in which pieces are *observed*. Winners can only be declared at the end of an observation phase. - **Observed piece:** An observed piece is drawn as a $\triangle$ or $\square$ symbol, or placed as a token, in a square on the board. Observed pieces can not be further interacted with, nor can the square in which it sits. - **Initiating player:** The last player to go before the observation phase is said to have *initiated* the observation phase. - **Mode of observation:** At higher levels, there are two modes of observation: The *standard* mode and the *rotated* mode, where in the latter $\triangle$ (resp. $\square$) states and $+$ (resp. $-$) states switches roles. The mode of observation is chosen by the player initiating the observation phase *before* rolling for values but *after* rolling for positions (locating empty squares). - **View:** At Level 5, it becomes possible for a player to change the *view* of the board on their turn, choosing between the *standard* view and the *rotated* view. The latter changes the dice by switching $\triangle$ (resp. $\square$) states for $+$ (resp. $-$), and vice versa. This opens up for interacting with $+$ and $-$ states, while simultaneously blocking $\triangle$ and $\square$ states from interaction. - **View token:** A coin-shaped token with $\triangle/\square$ on one side and $+/-$ on the other, used to keep track of which *view* the board is currently seen under (standard or rotated). ### Pieces Below is an overview of the pieces included in the game, together with which ones are needed at each level. ``` Player 1 Player 2 Level /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | 1-5 r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | 1-5 r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P r---r g---g b---b y---y p---p /\ /\ /\ /\ /\ | 4 | | 4 | | 4 | | 4 | | 4 | / 4\ / 4\ / 4\ / 4\ / 4\ 3-5 r---r g---g b---b y---y p---p R----R G----G B----B Y----Y P----P Bank /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | 2-5 r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | 2-5 r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P r---r g---g b---b y---y p---p /\ /\ /\ /\ /\ | 4 | | 4 | | 4 | | 4 | | 4 | / 4\ / 4\ / 4\ / 4\ / 4\ 5 r---r g---g b---b y---y p---p R----R G----G B----B Y----Y P----P View Token ___ ___ // \ // \ || △/□ | <-> || +/- | 5 \\___/ \\___/ ``` ### General Principles Following are the general principles common to every level of the game. 1. **One colour per turn:** On each turn, the player places either *one* or *two* dice of the *same colour* on the board, where the choice of colour is free as long as it has not already been used on one of the player's previous turns. 2. **One empty square per turn:** Each move must "spend" one empty square by placing a die one. 3. **One 4 or two 2s:** If the player places one die, then it must be placed in an empty square with the number $4$ up. If the player places two dice, then they must be placed in different squares, at least one of which is empty, with the numbers $\pm 2$ up. 4. **No self-interaction:** A player can not place a die in a square that already contains one of their own dice as part of their turn. 5. **Max two dice:** At no point in the game can a single square contain more than two dice. 6. **The observation phase:** When there are no more empty squares, the game enters the *observation phase*. The player who went last is said to have *initiated* the observation phase. 7. **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 with no winner declared, and there are no more empty squares on the board, then the game is a *tie*. 8. **Continuing the game:** If the observation phase concludes with no winner declared *and* there are empty squares left on the board, the game continues, with the turn going to the next player after the initaiting one. ### Moves Per Level Below is an overview of the moves and gameplay elements available at each level of the game. |Level/Element|Class. Move|Superpos. Move|Entang. Moves|Cancel. Moves|Rotated Mode|Rotated View|Post-obs. Interaction| |:---:|:------------:|:----------------:|:-------------:|:-------------:|:----------:|:----------:|:-:| |**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]]|✅|✅|✅|✅|✅|✅|✅|✅| ### The Classical Move The player places one die of their own shape on the board with the value $4$. ``` Player 1 Player 2 +--------------+ +--------------+ | | | | | /\ | | R---R | | / 4\ | | | 4 | | | r----r | | R---R | | | | | +--------------+ +--------------+ ``` 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. ``` Player 1 Player 2 +--------------+--------------+ +--------------+--------------+ | | | | | | | /\ | /\ | | R---R | R---R | | / 2\ | / 2\ | | | 2 | | | 2 | | | r----r | r----r | | R---R | R---R | | | | | | | +--------------+--------------+ +--------------+--------------+ ``` The superposition move adds one piece to the board. #### The Entangling Move The player places two dice of the same colour, of their own shape, with the value $2$ on the board, one in an empty square and the other in a square that contains exactly one die, belonging to the opponent. The game then adjusts itself by halving the value of the opponent's die, and placing another die of the same colour and (halved) value in the second square. ``` The Entangling Move Player 1, pre-adjustment +--------------+--------------+ | R---R | | | /\ | 4 | | /\ | | / 2\ R---R | / 2\ | Post-adjustment | r----r | r----r | +--------------+--------------+ | | | | R---R | R---R | +--------------+--------------+ | /\ | 2 | | /\ | 2 | | -> | / 2\ R---R | / 2\ R---R | Player 2, pre-adjustment | r----r | r----r | +--------------+--------------+ | | | | R---R | | +--------------+--------------+ | /\ | 2 | | R---R | | / 4\ R---R | | 2 | | | r----r | R---R | | | | +--------------+--------------+ ``` 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*. ``` The Half-Entangling Move Player 1, pre-adjustment +--------------+--------------+--------------+ | | R---R | | | R---R | /\ | 2 | | /\ | | | 2 | | / 2\ R---R | / 2\ | | R---R | r----r | r----r | | | | | +--------------+--------------+--------------+ Post-adjustment +--------------+--------------+--------------+ | | R---R | R---R | | R---R | /\ | 1 | | /\ | 1 | | -> | | 2 | | / 2\ R---R | / 2\ R---R | | R---R | r----r | r----r | | | | | +--------------+--------------+--------------+ Player 2, pre-adjustment +--------------+--------------+--------------+ | | R---R | | | /\ | /\ | 2 | | R---R | | / 2\ | / 2\ R---R | | 2 | | | r----r | r----r | R---R | | | | | +--------------+--------------+--------------+ Post-adjustment +--------------+--------------+--------------+ | | R---R | R---R | | /\ | /\ | 2 | | /\ | 2 | | -> | / 2\ | / 1\ R---R | / 1\ R---R | | r----r | r----r | r----r | | | | | +--------------+--------------+--------------+ ``` The (half-)entangling move adds one piece to the board. #### The Cancelling Move The player places two dice of the same colour on the board: One of the opponent's shape with the value $-2$, placed in a square occupied by the opponent, and one of their own shape in an empty square, with the value $2$ (Player 1) resp. $-2$ (Player 2). (The latter minus sign has no effect at Level 3 where the move is introduced, but becomes crucial from Level 4 onwards.) The game then adjusts itself by halving the value of the opponent's die, and placing another die of the same shape and colour and (halved) value in the second square. ``` The Cancelling Move Player 1, pre-adjustment Post-adjustment +--------------+--------------+ +--------------+--------------+ | r---r | | | r---r | R---R | | R---R |-2 | | /\ | | R---R |-2 | | /\ | 2 | | | | 4 | r---r | / 2\ | -> | | 2 | r---r | / 2\ R---R | | R---R | r----r | | R---R | r----r | | | | | | | +--------------+--------------+ +--------------+--------------+ Player 2, pre-adjustment Post-adjustment +--------------+--------------+ +--------------+--------------+ | /\ | | | /\ | R---R | | /\ /-2\ | R---R | | /\ /-2\ | /\ |-2 | | | / 4\ R----R| |-2 | | -> | / 2\ R----R| / 2\ R---R | | r----r | R---R | | r----r | r----r | | | | | | | +--------------+--------------+ +--------------+--------------+ ``` 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*. ``` The Half-Cancelling Move Player 1, pre-adjustment +--------------+--------------+--------------+ | | r---r | | | R---R | R---R |-2 | | /\ | | | 2 | | | 2 | r---r | / 2\ | | R---R | R---R | r----r | | | | | +--------------+--------------+--------------+ Post-adjustment +--------------+--------------+--------------+ | | r---r | R---R | | R---R | R---R |-2 | | /\ | 1 | | -> | | 2 | | | 1 | r---r | / 2\ R---R | | R---R | R---R | r----r | | | | | +--------------+--------------+--------------+ Player 2, pre-adjustment +--------------+--------------+--------------+ | | /\ | | | /\ | /\ /-2\ | R---R | | / 2\ | / 2\ R----R| |-2 | | | r----r | r----r | R---R | | | | | +--------------+--------------+--------------+ Post-adjustment +--------------+--------------+--------------+ | | /\ | R---R | | /\ | /\ /-2\ | /\ |-2 | | -> | / 2\ | / 1\ R----R| / 1\ R---R | | r----r | r----r | r----r | | | | | +--------------+--------------+--------------+ ``` The (half-)cancelling move **does not add any pieces to the board**. ### The Observation Phase With a few exceptions covered below, all rolls are made in the following way: Players take turns to choose a square containing one of their dice and ask, "is this square mine?" Then they roll a die. If the result is an even number, this is interpreted as *yes*, and if the result is an odd number, it is interpreted as a *no*. Update the board accordingly, placing symbols and updating affected states, and repeat until there are no more dice on the board. #### Procedure at Higher Levels At lower levels, the order in which the board is observed does not matter, but at Level 4 and Level 5 the below procedure should be followed: - **Step 1 (Level 5 only):** Set the *view* to *standard*. - **Step 2:** Roll for positions to determine which squares are empty and which are filled. - Remove the dice from any cancelled squares - Roll for (half-)superposition states - Roll for half-cancelled states - **Step 3:** Let the initiating player choose a *mode of observation*, and roll for symbols. - Option 1: Standard mode - Replace classical $\triangle$ and $\square$ states by $\triangle$ and $\square$ symbols without rolling - Roll for *plus* and *minus* states (independently) - Roll for *entangled* states (interdependently) - Option 2: Rotated mode - Replace *plus* and *minus* states by $\triangle$ and $\square$ symbols without rolling - Roll for *classical* states (independently) - Roll for *entangled* states (interdependently) - **Step 4:** Declare a winner if there is one, or a tie if there is no winner and no empty squares, or else continue. - **Step 5 (Level 5 only)**: Replace all observed $\triangle$ and $\square$ symbols with classical $\triangle$ and $\square$ states. After the observation phase, if no winner or tie was declared, the turn goes to the non-initiating player. #### How to Roll Use the tables below to roll for the outcomes of each state. There are two tables: One to roll for positions (as in Step 2 of the above procedure), and one to roll for symbols (as in Step 3). For easy reference, the square configurations referred to below are listed at the end of the chapter. ##### Rolling for Positions |State|Possible outcomes|Pr(Non-empty)|Non-empty if d6 rolls...|Adjust other squares based on outcome?| |:---:|:--:|:--:|:-:|:-:| |Superposition square|Classical state, Empty|$50\%$|Even number|Yes| |Cancelled square|Empty|$0\%$|-|No| |Half-cancelled square|Classical state, Empty|$3\%$|Snake-eyes (roll twice)|Yes| |Half-cancelling move, uninteracted square|Classical state, Empty|$2/5 = 40\%$|$1$ or $2$ (roll again on $6$)|Yes| |Plus/minus superposition|Plus/minus state, Empty|$50\%$|Even number|Yes| For half-entangled states, roll to see which of the two halves the original superposition resolves to, then update the board accordingly. For twice half-cancelled squares, first check that both cancellations succeeded, then roll to see which of the two target squares the superposition resolves to (probability 50/50). ##### Rolling for Symbols, Player 1 |State|Mode|Pr($\triangle$)|$\triangle$ if d6 rolls...|Adjust other squares based on outcome?| |:---:|:--:|:--:|:--:|:-:| |Classical|Standard|$100\%$|-|No| | |Rotated|$50\%$|Even number|No| |Entangled|Either|$50\%$|Even number|Yes| |Plus|Standard|$50\%$|Even number|No| | |Rotated|$100\%$|-|No| |Half-cancelling move, target square|Standard|$2/3 = 66\%$|Not $1$ or $2$|No| | |Rotated|$97\%$|Not snake-eyes (roll twice)|No| ##### Rolling for Symbols, Player 2 |State|Mode|Pr($\square$)|$\square$ if d6 rolls...|Adjust other squares based on outcome?| |:---:|:--:|:--:|:--:|:-:| |Classical|Standard|$100\%$|-|No| | |Rotated|$50\%$|Even number|No| |Entangled|Either|$50\%$|Even number|Yes| |Minus|Standard|$50\%$|Even number|No| | |Rotated|$100\%$|-|No| |Half-cancelling move, target square|Standard|$2/3 = 66\%$|Not $1$ or $2$|No| | |Rotated|$97\%$|Not snake-eyes (roll twice)|No| ### Switching the View Flip the *view token*, ``` Standard view Rotated view ___ ___ // \ // \ || △/□ | <-> || +/- | \\___/ \\___/ ``` then implement the following changes: ``` Plus P1 classical +--------------+ +--------------+ | R---R | | | | /\ | 2 | | | /\ | | / 2\ R---R | -> | / 4\ | | r----r | | r----r | | | | | +--------------+ +--------------+ Minus P2 classical +--------------+ +--------------+ | R---R | | | | /\ |-2 | | | R---R | | / 2\ R---R | -> | | 4 | | | r----r | | R---R | | | | | +--------------+ +--------------+ P1 classical P1 Plus +--------------+ +--------------+ | | | r---r | | /\ | | /\ | 2 | | | / 4\ | <-> | / 2\ r---r | | r----r | | r----r | | | | | +--------------+ +--------------+ P2 classical P2 Minus +--------------+ +--------------+ | | | R---R | | R---R | | /\ |-2 | | | | 4 | | <-> | / 2\ R---R | | R---R | | R----R | | | | | +--------------+ +--------------+ P1 superposition P1 plus-superposition +--------------+--------------+ +--------------+--------------+ | | | | r---r | r---r | | /\ | /\ | | /\ | 1 | | /\ | 1 | | | / 2\ | / 2\ | <-> | / 1\ r---r | / 1\ r---r | | r----r | r----r | | r----r | r----r | | | | | | | +--------------+--------------+ +--------------+--------------+ P2 superposition P2 minus-superposition +--------------+--------------+ +--------------+--------------+ | | | | R---R | R---R | | R---R | R---R | | /\ |-1 | | /\ |-1 | | | | 2 | | | 2 | | <-> | / 1\ R---R | / 1\ R---R | | R---R | R---R | | R----R | R----R | | | | | | | +--------------+--------------+ +--------------+--------------+ Entangled and cancelled squares: Do nothing +--------------+--------------+ +--------------+ +--------------+ | R---R | R---R | | /\ | | r---r | | /\ | 2 | | /\ | 2 | | | /\ /-2\ | | R---R |-2 | | | / 2\ R---R | / 2\ R---R | | / 2\ R----R| | | 2 | r---r | | r----r | r----r | | r----r | | R---R | | | | | | | | +--------------+--------------+ +--------------+ +--------------+ Half-interacted: Update uninteracted half only Example: P2 half-cancelled +--------------+--------------+ +--------------+--------------+ | r---r | | | r---r | R---R | | R---R |-2 | | R---R | | R---R |-2 | | /\ |-1 | | | | 1 | r---r | | 2 | | | | 1 | r---r | / 1\ R---R | | R---R | R---R | | R---R | R----R | | | | | | | +--------------+--------------+ <-> +--------------+--------------+ | R---R | | | R---R | | | /\ | 1 | | | | /\ | 1 | | | | / 2\ R---R | | | / 2\ R---R | | | r----r | | | r----r | | | | | | | | +--------------+--------------+ +--------------+--------------+ ``` ## Possible Square Configurations ``` P1 classical state P2 classical state +--------------+ +--------------+ | | | | | /\ | | R---R | | / 4\ | | | 4 | | | r----r | | R---R | | | | | +--------------+ +--------------+ P1 superposition P2 superposition +--------------+ +--------------+ | | | | | /\ | | R---R | | / 2\ | | | 2 | | | r----r | | R---R | | | | | +--------------+ +--------------+ Entangled P1 half-entangled P2 half-entangled +--------------+ +--------------+ +--------------+ | R---R | | R---R | | R---R | | /\ | 2 | | | /\ | 1 | | | /\ | 2 | | | / 2\ R---R | | / 2\ R---R | | / 1\ R---R | | r----r | | r----r | | r----r | | | | | | | +--------------+ +--------------+ +--------------+ (With partner square) (With partner square) (With partner square) P1-cancelled square P2-cancelled square +--------------+ +--------------+ | r---r | | /\ | | R---R |-2 | | | /\ /-2\ | | | 2 | r---r | | / 2\ R----R| | R---R | | r----r | | | | | +--------------+ +--------------+ Plus state Minus state +--------------+ +--------------+ | R---R | | R---R | | /\ | 2 | | | /\ |-2 | | | / 2\ R---R | | / 2\ R---R | | r----r | | r----r | | | | | +--------------+ +--------------+ (No partner square) P1 half-cancelling move, Half-cancelled square Target square Uninteracted square +--------------+ +--------------+ +--------------+ | r---r | | R---R | | | | R---R |-2 | | | /\ | 1 | | | R---R | | | 1 | r---r | | / 2\ R---R | | | 2 | | | R---R | | r----r | | R---R | | | | | | | +--------------+ +--------------+ +--------------+ (No partner square) P2 half-cancelling move, Half-cancelled square Target square Uninteracted square +--------------+ +--------------+ +--------------+ | /\ | | R---R | | | | /\ /-2\ | | /\ |-2 | | | /\ | | / 1\ R----R| | / 1\ R---R | | / 2\ | | r----r | | r----r | | r----r | | | | | | | +--------------+ +--------------+ +--------------+ Plus-superposition Minus-superposition +--------------+ +--------------+ | r---r | | R---R | | /\ | 1 | | | /\ | 1 | | | / 1\ r---r | | / 1\ R---R | | r----r | | R----R | | | | | +--------------+ +--------------+ (With partner square) (With partner square) ``` [^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).