## Welcome! Welcome to quantum tic-tac-toe, or *TiqTaqToe*! You are about to embark on a journey into the strange and fascinating world of quantum mechanics, a set of laws governing the smallest particles in our universe—and thus, in a way, *all* our universe, including you and me! TiqTaqToe expands upon the age-old game of "Naughts & Crosses" by introducing new moves to the game, as made possible by the laws of quantum physics. By introducing one new move at a time, we get five different versions of the game, starting with the beginner-friendly [[TiqTaqToe with Superpositions (d4s)]], and ending with the mind-bending [[Fully Quantum TiqTaqToe (d4s)]]. Each version comes with its own set of strategies, and a superb move in one version might turn out to be inferior in the next. Along the way, each new move will illustrate one new concept central to quantum physics, and the probabilities of each outcome will be consistent with the real quantum physical laws governing our nature. The curious player will then be invited to check out [[The Physics of TiqTaqToe]], where you will learn more about how the moves you perform connect to the scientific breakthroughs of the past century, as well as the quantum technologies of tomorrow! Here is an overview of the five levels of TiqTaqToe (with "Level 0" as the classical game): %% ``` Level 5 – Master: Fully Quantum TiqTaqToe | Level 4 – Expert: TiqTaqToe with Bases | Level 3 – Advanced: TiqTaqToe with Interference | Level 2 – Adept: TiqTaqToe with Entanglement | Level 1 – Beginner: TiqTaqToe with Superpositions | (Level 0 – Novice: Classical Tic-Tac-Toe) ``` %% - **Level 1 – Beginner: [[TiqTaqToe with Superpositions (d4s)]]** - **Level 2 – Adept: [[TiqTaqToe with Entanglement (d4s)]]** - **Level 3 – Advanced: [[TiqTaqToe with Interference (d4s)]]** - **Level 4 – Expert: [[TiqTaqToe with Bases (d4s)]]** - **Level 5 – Master: [[Fully Quantum TiqTaqToe (d4s)]]** Each level adds one new element to the one before it, which leads to a more challenging game with a greater set of possible moves and outcomes. And yet, in the end, the winner is still the player who gets a three-in-a-row first! But first, let us introduce the pieces. ### The TiqTaqToe Dice ![[IMG_0136.jpeg|400]] TiqTaqToe comes with $40$ d4 dice—that is, dice with four numbers each—which will be placed on the classic nine-square three-in-a-row board.[^1] The dice come in ten colours, five for each player.[^2] In the set pictured above, five sets of dice have white numbers and five sets have gold numbers (except for the red and blue numbers), which makes it easy to distinguish which dice belong to which player: Player 1 gets the white-numbered dice, and Player 2 gets the gold-numbered dice. As you can see, there are four dice per colour, two of which are "normal" (that is, of mainly white or yellow numbering), while two are "special": one with mainly red numbers, and one with mainly blue numbers. In addition are two d6 dice, one for each player,[^3] where the colour of the numbering matches that of the d4 dice belonging to the player (white/gold). Aside from its function in-game, this makes it easy for the players to recall which colour (white/gold) belongs to them. Finally, we have the *Observation Token*, pictured on the left, which has an "XO" symbol on one side and a "+-" symbol on the other. ![[IMG_0138 1.jpeg|150]] ![[IMG_0139 1.jpeg|150]] Not all the pieces are needed in every level, though. How the pieces should be distributed are explained in the chapter for each level. Additionally, the [[TiqTaqToe Player Reference (d4s)]] contains an overview of who should have what pieces in each level. As for the game board, as mentioned the game is played on a $3 \times 3$ grid. This can be done using pen and paper, or a pencil and eraser in place of a pen for less paper waste. Alternatively, a small, portable whiteboard, together with a dry-erase marker, makes for an excellent playing board![^4] ### Main Rules The various versions have their own movesets, but the main rules are common to all versions, listed below for easy reference. Certain details won't become relevant until later, so don't worry too much about them in the beginning—but if you ever find yourself wondering why a particular move is or is not allowed, you might find the answer in this list. 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. **No self-entanglement:** A player can not place a die in a square that already contains one of their own dice as part of their turn. 4. **Max two dice per square:** At no point in the game can a square contain more than two d4 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*. These rules, together with a terms glossary and an overview of the moves, can be found in [[TiqTaqToe Player Reference (d4s)]]. For more on the physical principles that underlie the rules of TiqTaqToe, including a mathematical treatment of the game, see [[The Physics of TiqTaqToe]]. ### The First Step With all of that out of the way, it is time to embark on our journey into the quantum realm. So when you are ready, take the first step, and **enter Level 1: [[TiqTaqToe with Superpositions]]!** [^1]: For those new to d4 dice, they are pyramid-shaped dice, where the number pointing upwards gives the result. (Some d4 dice have the number written along the bottom of each side instead, so that it is the number placed *lowest* that is the number rolled.) [^2]: Nine colours would also suffice: Five colours to the first player, and four colours to the second, leading to 36 dice in total. However, five colours per player makes it easier to alternate who starts. [^3]: The game can also be played with just one d6 die. [^4]: One could also improve replayability by making nine $X$/$O$ tokens (one side $X$ and the other $O$). If so, ensure that every token comes with both symbols, since higher levels allow boards that are *all* $X$ or *all* $O$. [^5]: In the manufactured version these will be $1$s.