## Level 1 (Classic): Superpositions and Entanglement Tic-tac-toe, or three-in-a-row, naughts and crosses, or bondesjakk ("farmer's chess") in Norwegian, dates back [at least three thousand years](https://en.wikipedia.org/wiki/Tic-tac-toe#History). Its rules are familiar to most from childhood: Each player takes turns placing symbols (traditionally $X$s and $O$s) on a 3 by 3 grid, and the first player to get three in a row—horizontally, vertically or diagonally—is declared the winner. If no player has three in a row by the time that all nine squares have been filled, the game ends in a tie. However, a classic though it is, tic-tac-toe is of limited interest as a game of strategy. This is because it is a *solved* game: An optimal strategy exists, and as long as both players play optimally, the game is guaranteed to end in a tie. What's more, the optimal strategy is pretty easy to discover, and essentially comes down to both players blocking the opponent's chance for a three-in-a-row at every opportunity.[^11] With a few quantum moves, TiqTaqToe changes this situation. At this level, we introduce two such moves: The *superposition* move and the *entangling* move. At this level, each player should have $4$ dice of each colour of their own shape, as well as a normal d6 die to roll for outcomes. The extra die of the opponents shape will not come into play until the next level. ``` Player 1 Player 2 /\ /\ /\ /\ /\ R---R G---G B---B Y---Y P---P / 4\ / 4\ / 4\ / 4\ / 4\ | 4 | | 4 | | 4 | | 4 | | 4 | 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 | 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 | 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 | r----r g----g b----b y----y p----p R---R G---G B---B Y---Y P---P ``` ##### Table of Contents - Phase 1: Taking Turns - The Classical Move - The Superposition Move - The Entangling Move - Entangling Superpositions - Phase 2: The Observation Phase - Phase 3: Deciding the Game - Observation Phase Example - Balance and Strategy - Levelling Up ### Phase 1: Taking Turns On each turn the player has (up to) three options: Do they make a classical move to try to capture an empty square, a superposition move to fill two spaces at once, or an entangling move to disrupt the opponent's placement? #### The Classical Move To make a classical move, place one die of your own shape ($\triangle$ for Player X, $\square$ for Player O) in an empty square with the number $4$ up.[^13] ``` +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | | | | | /\ | | | | / 4\ | | | | r----r | | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` Intuitively, this means that "all" of that colour is present in that square; or in other words, that there is a $4$ out of $4 = 100\%$ probability that the player's first symbol, or *piece* (see [[TiqTaqToe Player Reference]] for a terms glossary), would be observed in that square. #### The Superposition Move To make a superposition move, place two dice of the same colour with the numbers $2$ facing up, each in an empty square on the board. ``` +--------------+--------------+--------------+ | | | | | | | /\ | | | | / 2\ | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 2\ | | | | r----r | | | | | | | +--------------+--------------+--------------+ ``` The $2$s show that "half" the colour is present in each square; or in other words, that there is a two-fourths, or $50\%$, probability that the piece would be observed to be in either square. (And of course, once the piece is observed to be in one square, we will also know that it is definitely not in the other.) Hence, this move lets the player place their piece in a superposition of "two places at once", such that its position will only be decided once it is observed, which happens in the observation phase. #### The Entangling Move The entangling move allows a player to disrupt a move that the other player already made. Say the first player performed a classical move in the middle square. Then the second player can place two dice of the same colour with the number $2$ up in two different squares, just like they did for the superposition move. The difference is that now, the player lets one of its dice *invade* (so to speak) the square of the opponent. ``` +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | R---R | R---R | | | /\ | 2 | | | 2 | | | | / 4\ R---R | R---R | | | r----r | | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` The other die must be placed in an empty square, as per the general rule that every move must consume at least one empty square. But now something looks off: If we interpret the numbers, the dice are currently telling us that there is a $\triangle$ in the middle with $100\%$ certainty, *and* a $\square$ in the middle with $50\%$ probability. Well, it can't be both! The board resolves this by automatically adjusting itself after an entangling move: The $4$ in the middle is *split in halves*, where the die in the original square goes from $4$ to $2$, and a new die of the same colour is brought to the second square, also with the number $2$. ``` +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | R---R | R---R | | | /\ | 2 | | /\ | 2 | | | | / 2\ R---R | / 2\ R---R | | | r----r | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` One way to remember this is to imagine that Player 2 "invades" Player 1's square and "drags" half their probability with them over to another square of their choosing. As for the superposition move, the choice of the second square is free. ``` +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 2\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | R---R | | | | /\ | 2 | | | | | / 2\ R---R | | | | r----r | | | | | | | +--------------+--------------+--------------+ ``` With this, there are two possible, equally probable outcomes: Either $\triangle$ is in the first square and $\square$ is in the second square, *or* $\square$ is in the first square and $\triangle$ is in the second square: Learning where $\triangle$ is immediately reveals where $\square$ is (namely the other square), so we will only have to roll for the position of one piece to decide the position of both of them. —And that is exactly what it means for two objects to be entangled![^1] #### Entangling Superpositions The same entangling move can be used to attack a superposition move. Let's say Player 1 started with the following superposition move. ``` Move 1 +--------------+--------------+--------------+ | | | | | | | /\ | | | | / 2\ | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 2\ | | | | r----r | | | | | | | +--------------+--------------+--------------+ ``` Player 2 makes an entangling move, opting to place the other die in the empty middle-right square. ``` Move 2, pre-adjustment +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 2\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | R---R | | | | | 2 | | | | | R---R | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 2\ | | | | r----r | | | | | | | +--------------+--------------+--------------+ ``` Now the game adjusts itself: Player 1's $2$ splits in half, this time into $1$s, with one $1$ remaining in place and the other going to the other square. ``` Move 2, post-adjustment +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 2\ | | | | r----r | | | | | | | +--------------+--------------+--------------+ ``` Just like before, your piece "invaded" the square of the opponent and "dragged" half their probability weight over to another square. There is now a $1$ in $4 = 25\%$ probability that Player 1's piece will end up in either the middle-right or the top right square in the end. The result is a "half-entangling move": You have entangled with one half of a superposition, while the other half of the superposition remains untouched. There are thus three possible outcomes: *If* the opponent's piece is later found to be in the unentangled half (which happens with $50\%$ probability), *then* your piece is just in a normal superposition over otherwise empty squares, and there is no entanglement anymore. ``` +--------------+--------------+--------------+ | | | | | | | R---R | | | | | 2 | | | | | R---R | | | | | +--------------+--------------+--------------+ | | | | | | | R---R | | | | | 2 | | | | | R---R | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / \ | | | | +----+ | | | | | | | +--------------+--------------+--------------+ ``` If it is found to be in the entangled half of its original superposition, then the situation ends up being as if you had performed a normal entangling move. ``` +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 2\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 2\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` But we are getting ahead of ourselves. Back in our example game, there is nothing stopping you from later *also* entangling with the other half the opponent's superposition. Here, Player 1 played a classical move in the lower right corner, making it your turn once again. ``` Move 3 +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | | | | | /\ | | /\ | | / 2\ | | / 4\ | | r----r | | g----g | | | | | +--------------+--------------+--------------+ Move 4, pre-adjustment +--------------+--------------+--------------+ | | | R---R | | G---G | | /\ | 2 | | | | 2 | | | / 1\ R---R | | G---G | | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | G---G | | | | /\ | 2 | | | /\ | | / 2\ G---G | | / 4\ | | r----r | | g----g | | | | | +--------------+--------------+--------------+ Move 4, post-adjustment +--------------+--------------+--------------+ | G---G | | R---R | | /\ | 2 | | | /\ | 2 | | | / 1\ G---G | | / 1\ R---R | | r----r | | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | | /\ | 2 | | | | | / 1\ R---R | | | | r----r | | | | | +--------------+--------------+--------------+ | G---G | | | | /\ | 2 | | | /\ | | / 1\ G---G | | / 4\ | | r----r | | g----g | | | | | +--------------+--------------+--------------+ ``` Note how the opponent's red piece is now maximally spread out over the board: Their red dice are all $1$s, meaning the piece has a $25\%$ probability of being in either of those four squares. This is as far as it goes in terms of splitting: Even if we wanted to, we wouldn't be able to split again, since the "no self-entanglement" rule stops us from placing a die in an already-entangled square. ### Phase 2: The Observation Phase When there are no more empty squares left, the game enters the *observation phase*. Here's the procedure: 1. The last player to go before the board was full is said to have *initiated* the observation. 2. For each classical piece (square containing a single die with the number $4$), remove the die and draw/place the corresponding symbol/token in the square: X (or $\triangle$) for Player 1 and O (or $\square$) for Player 2. 3. Then take turns rolling for the remaining pieces: Choose a square where you have a die with the number $2$ up, point at it and ask "is this square mine?", then choose how to interpret the roll of the (normal) d6. For example, "even number means yes, odd number means no". Then roll, fill in the square if it was found to be there, and adjust the board accordingly by removing the other die or dice of the same colour, or if it was not there, adjust the probability of the other square, filling it in with your symbol if it becomes $100\%$. 4. When there are no more dice on the board, *decide* the game. Note that the order in which you make the rolls does not matter.[^9] Also, the procedure above deliberately avoids rolling for dice with the number $1$ up ($25\%$ probability), as through playtesting we have found it much easier to avoid confusion by focusing on the $2$-numbered dice, and rolling for those until there is nothing left to roll. This is possible because as the $2$-dice are resolved, the $1$-dice will either double to become $2$s themselves, or disappear from the board. ### Phase 3: Deciding the Game After completing the observation phase, and *only* then, the game is decided, according to which of four possible cases the board falls into: 1. Exactly one player has a three-in-a-row. That player is wins the game. 2. *Both* players have threes-in-a-row. The game is a tie.[^14] 3. Neither player has three-in-a-row, and there are no more empty spaces left on the board. The game is a tie. 4. Neither player has three-in-a-row, and there are still empty spaces left on the board. The game continues, with the turn going to the next player after the initiating player. In case 4, the game continues as before, with the exception that filled-in squares are off-limits. When there are no empty squares left, then the game enters the observation phase a second time, and so on until the game is decided. ### Observation Phase Example Let's look at an example of how two players might carry out the observation phase. Consider the following board: ``` +--------------+--------------+--------------+ | G---G | B---B | R---R | | /\ | 2 | | /\ | 2 | | /\ | 2 | | | / 1\ G---G | / 2\ B---B | / 1\ R---R | | r----r | b----b | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | Y---Y | /\ | /\ | 2 | | | | 2 | | / 4\ | / 1\ R---R | | Y---Y | g----g | r----r | | | | | +--------------+--------------+--------------+ | G---G | B---B | | | /\ | 2 | | /\ | 2 | | Y---Y | | / 1\ G---G | / 2\ B---B | | 2 | | | r----r | b----b | Y---Y | | | | | +--------------+--------------+--------------+ ``` We begin by replacing all the pieces in a classical state ($4$-dice) with their respective symbols/tokens. ``` +--------------+--------------+--------------+ | G---G | B---B | R---R | | /\ | 2 | | /\ | 2 | | /\ | 2 | | | / 1\ G---G | / 2\ B---B | / 1\ R---R | | r----r | b----b | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | Y---Y | /\ | /\ | 2 | | | | 2 | | / \ | / 1\ R---R | | Y---Y | +----+ | r----r | | | | | +--------------+--------------+--------------+ | G---G | B---B | | | /\ | 2 | | /\ | 2 | | Y---Y | | / 1\ G---G | / 2\ B---B | | 2 | | | r----r | b----b | Y---Y | | | | | +--------------+--------------+--------------+ ``` Then we take turns resolving each colour by pointing at a square with a $2$-die in it and asking, "Is this square mine?" and rolling the normal d6 die: The roll of an even number means the answer is "yes", and an odd numbers means "no". As a reminder, the order in which we choose to resolve the dice is irrelevant, since it is only at the end of the observation phase that a possible winner is declared. Let us start with the yellow d6 dice in the middle-left and lower-right squares. This piece is the result of a superposition move, and it is not entangled with the opponent, so it is in either square with 50/50 probability, with the other square empty. Since the dice belong to Player 2, they point to the left square and ask, "Is this square mine?" They roll a $3$: The answer is *no*, so the piece must be in the lower-right. ``` +--------------+--------------+--------------+ | G---G | B---B | R---R | | /\ | 2 | | /\ | 2 | | /\ | 2 | | | / 1\ G---G | / 2\ B---B | / 1\ R---R | | r----r | b----b | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | /\ | /\ | 2 | | | | / \ | / 1\ R---R | | | +----+ | r----r | | | | | +--------------+--------------+--------------+ | G---G | B---B | | | /\ | 2 | | /\ | 2 | | +---+ | | / 1\ G---G | / 2\ B---B | | | | | r----r | b----b | +---+ | | | | | +--------------+--------------+--------------+ ``` Next, Player 1 wants to resolve their blue dice, placed in the top square and bottom square. These dice correspond to two entangled pieces, such that one piece is in one square and the other piece is in the other, again with 50/50 probability. Player 1 points to the top square and asks, "Is this square mine?" They roll a $6$, so the answer is *yes*: They place a $\triangle$ token in the top square, and Player 2's places a $\square$ token in the bottom square to represent the piece corresponding to their blue d6 dice. ``` +--------------+--------------+--------------+ | G---G | | R---R | | /\ | 2 | | /\ | /\ | 2 | | | / 1\ G---G | / \ | / 1\ R---R | | r----r | +----+ | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | /\ | /\ | 2 | | | | / \ | / 1\ R---R | | | +----+ | r----r | | | | | +--------------+--------------+--------------+ | G---G | | | | /\ | 2 | | +---+ | +---+ | | / 1\ G---G | | | | | | | | r----r | +---+ | +---+ | | | | | +--------------+--------------+--------------+ ``` The quantum state of Player 1's final piece, represented by the four red $1$-dice, has been spread over four squares after two half-entangling moves on the opponent's part. It is Player 2's turn to roll: They want to resolve their green dice, so they point to the bottom-left and ask, "Is this square mine?" They hope for a yes, since that will get them the bottom row, but they are out of luck: They roll a $1$, a *no*, and so their piece must be in the top-left square. ``` +--------------+--------------+--------------+ | | | R---R | | +---+ | /\ | /\ | 2 | | | | | | / \ | / 1\ R---R | | +---+ | +----+ | r----r | | | | | +--------------+--------------+--------------+ | | | R---R | | | /\ | /\ | 2 | | | | / \ | / 1\ R---R | | | +----+ | r----r | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / 2\ | | | | | | | | r----r | +---+ | +---+ | | | | | +--------------+--------------+--------------+ ``` Notice how Player 1's red die in the bottom left square has been changed from a $1$ to a $2$: Since Player 2's piece is now known to be in the top left square, we also know that Player 1's piece is *not* in the top left square. Since the half of the superposition that was entangled with Player 2's green dice had been spread over the top-left and bottom-left squares, with the top-right and middle-right parts of the superposition independent of the green dice, the result is that Player 1's piece has twice the probability of being observed on the bottom-left than it did before the roll. It is Player 1's turn to roll. Since the bottom-left square is the only square with a $2$-die, they point to it and ask, "Is this square mine?" They roll, and find that the answer is yes. ``` +--------------+--------------+--------------+ | | | | | +---+ | /\ | R---R | | | | | / \ | | 2 | | | +---+ | +----+ | R---R | | | | | +--------------+--------------+--------------+ | | | | | | /\ | R---R | | | / \ | | 2 | | | | +----+ | R---R | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / \ | | | | | | | | +----+ | +---+ | +---+ | | | | | +--------------+--------------+--------------+ ``` As you can see, this also means that it is *not* in either of the two squares on the right, so those dice are removed from the board. Player 2 is now left with a normal, unentangled superposition over the two squares. They resolve the final roll by pointing to the top-left square and asking, "Is this square mine?" They roll a $5$—no. ``` +--------------+--------------+--------------+ | | | | | +---+ | /\ | | | | | | / \ | | | +---+ | +----+ | | | | | | +--------------+--------------+--------------+ | | | | | | /\ | +---+ | | | / \ | | | | | | +----+ | +---+ | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / \ | | | | | | | | +----+ | +---+ | +---+ | | | | | +--------------+--------------+--------------+ ``` The game is undecided, and there are still empty squares left. That means that the game continues! The turn goes to the non-initiating player, which in this case means it is Player 1's turn. What would you do next? ### Balance and Strategy A natural question to ask for any two-player game is, is there a first-player advantage? On the one hand, Player 1 *can* place one more piece on the board than Player 2, which usually translates to a slightly higher win probability. On the other hand, in doing so they must also be placing the last piece on the board, giving the second player the opportunity to go first *after* the observation phase, should the game remain undecided. As we already saw in the previous level, this can be a very fortuitous position to be in! In fact, playtesting suggests that two skilled players will tend towards boards that gives each player close to exactly 50/50 probability of winning, not counting ties. This might for example be because the board they end up with is highly symmetrical, like the one below. ``` +--------------+--------------+--------------+ | | G---G | | | /\ | /\ | 2 | | R---R | | / 2\ | / 2\ G---G | | 2 | | | r----r | g----g | R---R | | | | | +--------------+--------------+--------------+ | B---B | B---B | | | /\ | 2 | | /\ | 2 | | /\ | | / 2\ B---B | / 2\ B---B | / 4\ | | b----b | b----b | y----y | | | | | +--------------+--------------+--------------+ | | G---G | | | /\ | /\ | 2 | | R---R | | / 2\ | / 2\ G---G | | 2 | | | r----r | g----g | R---R | | | | | +--------------+--------------+--------------+ ``` The trick, then, is to find ways to edge the probabilities in your favour at the expense of your opponent's. Whether optimal strategies to achieve this exist remains an open question, but anecdotal evidence suggests that something like a 70/30 advantage should be consistently achievable for a skilled player playing against a less experienced one. As for developing good strategies, I have found that it is usually easier to spot *bad* moves than good ones. In that sense, good play might simply come down to managing to avoid more bad moves than your opponent. Consider the following board. Do you see why placing a die in the right square would be a bad move for either player? If you were next, where would you place instead to maximize future chances for a three-in-a-row? (See the footnote for an answer.[^4]) ``` +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | R---R | R---R | | | /\ | 2 | | /\ | 2 | | | | / 2\ R---R | / 2\ R---R | | | r----r | r----r | | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` As another example, it turns out that a superposition move is a poor opener. Do you see why?[^5] ``` +--------------+--------------+--------------+ | | | | | | | /\ | | | | / 2\ | | | | g----g | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | | | | /\ | | | | / 2\ | | | | g----g | | | | | | | +--------------+--------------+--------------+ ``` ### Levelling Up The ruleset of this level was first developed by Evert van Nieuwenburg, and it is the version of the game that has seen the most play by far.[^2] It has the features of a classic, being easy to learn yet hard to master. Indeed, even young students seem to take to the game fast, and yet mastering *strategy* remains a challenge even for seasoned players! The game also provides a good illustration of the challenge involved in quantum reasoning, as even simple situations quickly become hard to reason about as the combinatorial tree explodes. And yet, these rules are still *really* only relying on classical conditional probabilities. The real weirdness of the quantum world comes from the fact that, unlike classical probabilities, quantum probabilities, or "amplitudes"[^8], can not only be negative, but even "in between" positive and negative![^10] As a consequence, quantum probabilities can *cancel out*, much like how two colliding waves can cancel out exactly to leave the water perfectly still. It is precisely this effect our next move exploits, so once you have played a few rounds using the current ruleset, it is time to graduate to **Level 2 (Advanced): [[TiqTaqToe with Interference]]!** [^11]: For an example of a still-solved variant of the game that *is* still interesting, because the optimal strategy is too complicated for humans to implement, see [Qubic](https://en.wikipedia.org/wiki/3D_tic-tac-toe#%22Qubic%22) [^13]:The players have free choice of which of their colours to use on each turn, but these ASCII diagrams always use the colours in the rainbow order: Red, green, blue, yellow, purple. Players are also encouraged to alternate going first, but here Player 1 will always be $\triangle$, corresponding to X. [^1]: If you are already familiar with basic probability theory, you may recognize this as mere conditional probabilities. Indeed, entanglement could be said to be the quantum generalization of conditional probabilities. Still, this move admittedly only involves only classical conditional probabilities rephrased in a quantum language. The *real* quantum stuff, that which has no classical probability analogue, comes in the next level. [^9]: For the mathy folks around here: You may want to convince yourself that a full board defines a probability distribution over classical tic-tac-toe boards, and that the observation phase is a procedure to sample a board from that distribution. (Hint: Adjusting the board between rolls is an a-posteriori renormalization of the marginal distribution.) [^14]: You may wonder why the game does not continue in this case. The answer is that at this level, if both players have threes-in-a-row, there is no way for one player to get an additional one; hence, the game is already decided. [^4]: The middle row is blocked for both players, since due to entanglement, the row will contain at least one of each piece, $\triangle$ and $\square$. Therefore, neither player has a chance of getting a three-in-a-row there, and the square should be avoided. By instead placing a piece in either the top left or bottom left corner (or both, using a superposition move), you open up for three-in-a-rows on the diagonals and along the left edge, which gives you a shot at winning regardless of where the entangled pieces on the middle row land. [^5]: The middle square is the strongest position on the board, so naturally both players want it. If player 1 opens with a superposition move, regardless of whether they placed one half in the middle square, Player 2 can now ensure that Player 1 only gets a $25\%$ chance of capturing it, while giving themselves a $50\%$ probability of capturing it, using the half-entangling move. [^2]: A version of this game is playable on [tiqtaqtoe.com](https://www.tiqtaqtoe.com) as "Level 3 – Triple entanglement", with the differences that players can take a turn to measure a square instead of placing a new piece, and that entangling both halves of a superposition is disallowed. [^8]: In reference to wave mechanics; see [[The Physics of TiqTaqToe]]. [^10]: Such numbers are known as "imaginary" or "complex" numbers. They won't show up in this game, but they are essential to the mechanics underlying [Quantum Chess](https://quantumchess.fandom.com/wiki/Quantum_Chess_Wiki).