God does not play dice. *– Albert Einstein, 1926* ## Welcome to TiqTaqToe! 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 that make up our universe—and thus, the laws governing *all* our universe, including you and me! TiqTaqToe expands upon the age-old game of three-in-a-row by introducing new moves to the game, as made possible by the laws of quantum mechanics. The base level of the game starts by adding *two* new, uniquely quantum moves to classical tic-tac-toe: The superposition move and the entangling move. Then the second and third levels each add one move to the mix that illustrates a uniquely quantum effect: *Phase cancellation* (a.k.a. *interference*) and *basis change*. With each new move, much of the strategy gets turned on its head, and a superb move in one level might spell doom in the next. For each level, we also get a conceptually more challenging ruleset. Players will find their efforts rewarded, however, as each level gradually opens for more strategic play, making for a progressively deeper and more rewarding experience. But one rule that is the same for all levels is that once the board is full, it is *observed*, at which point we roll dice to see how the quantum states "collapse" to become the familiar X and O symbols. Then, if one player gets a three-in-a-row, that player wins! Otherwise, the game continues. The act of observing is carried out by rolling dice, according to probabilities derived using real quantum theory. (The curious player can check out [[The Phenomena Behind TiqTaqToe]] to learn more about how each move connects to the major scientific revolution of the past century—and the quantum technologies of tomorrow.) TiqTaqToe has one three main levels: - **[[TiqTaqToe with Entanglement|Level 1 (Classic): Superpositions and Entanglement]]** - **[[TiqTaqToe with Interference|Level 2 (Advanced): Interference]]** - **[[TiqTaqToe with Complementarity|Level 3 (Expert): Complementarity]]** There is also a [[TiqTaqToe with Superpositions|Level 0 (Novice): Superpositions]], which is like Level 1 without the entangling move. This "zeroth" level is useful as a gentle introduction for younger players, but is of less interest in its own right, as over the course of a few games one quickly comes to realize that Player 1 can always force a win. ### Main Rules While different levels have different moves available, the main rules of the game are common to all of them. Certain careful formulations won't become relevant until higher levels, so don't worry too much about minute details while learning the game—but if you ever find yourself wondering why a particular move is or is not allowed, you might find your answer below. 1. **One 4 or two 2s:** On each turn, the player places either one $4$ or two $2$s (one of which may for certain moves be negative). 2. **One colour per turn:** On each turn, the player places dice of one colour that the player did not already use on a previous turn. 3. **No self-interaction:** A player can not place a die in a square that already contains a die of their own. As a corollary, a square may house at most two dice (one per player). 4. **One empty square per turn:** On each turn, at least one previously-empty square must end up filled. 5. **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 (sometimes referred to as "the initiating player"). 6. **Deciding the game:** If one player has more threes-in-a-row than the other player at the end of the observation phase, then that player is declared the winner. If not, and the board is full, then the game is a tie. 7. **Continuing the game:** If there are no winners at the end of the observation phase and there are empty squares remaining, then the game continues, with the turn going to the next player after the initaiting player. These rules can also be found in [[TiqTaqToe Player Reference]], together with a terms glossary and an overview of the moves, [[The Pieces of TiqTaqToe|With all that out of the way, let us start by introducing the player pieces!]] ### Table of Contents ##### Main Manual 1. Introduction: [[The Pieces of TiqTaqToe]] 2. Level 1 (Classic): [[TiqTaqToe with Entanglement]] 3. Level 2 (Advanced): [[TiqTaqToe with Interference]] 4. Level 3 (Expert): [[TiqTaqToe with Complementarity]] ##### Appendices - [[TiqTaqToe Player Reference]] - [[The Phenomena Behind TiqTaqToe]] - [[The Mathematics of TiqTaqToe]] - Level 0 (Novice): [[TiqTaqToe with Superpositions]] ### History etc. Evert etc., tiqtaqtoe.com, [[TiqTaqToe with Superpositions]] What is entanglement? In short, it is *interdependent superpositions*, meaning superpositions over two interdependent *pieces*, such that the *state* can not be described without talking about both pieces at once.[^6] For example, consider the situation "If piece A is at position 1, then piece B is at position 2, and if piece A is at position 2, then piece B is at position 1". Note the interdependence: If the superposition states were *independent*, then the statement would instead be that "piece A is either at position 1 or 2, and piece B is either at position 1 or 2", meaning that both pieces could end up at position 1 or at position 2. Instead, we have that the position of piece B *determines* the position of piece A, and vice versa.