## Level 0 (Novice): Superpositions Tic-tac-toe, or three-in-a-row, 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.[^1] With a few quantum moves, TiqTaqToe changes this situation. The first such quantum move is the *superposition* move, and leads to the first version of the game, *TiqTaqToe with Superposition*.[^2] ### Distributing the Pieces TiqTaqToe is played using dice: Four-sided dice (d4s) for Player 1 and six-sided dice (d6s) for Player 2. The dice are not rolled, and they have different numbering from standard dice—these numbers will represent the *probability* of various outcomes. Each player has dice of five colours, and there are four dice of each colour, in addition to one die of the opponent's shape (a d6 for Player 1, and a d4 for Player 2). For this version though, we will only need two dice per colour of the usual shape. ``` 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 ``` You will also need a standard six-sided die, to be used to roll for the outcomes. If you have two, give one to each player. As explained in [[The Road to TiqTaqToe Mastery]] (see also [[TiqTaqToe Player Reference]]), for each turn, the player places dice on the board of a colour that has not already been used. At this level, there are two moves available to the player: The classical move and the superposition move. ### Move 1: The Classical Move When making a classical move, the player places *one* die of their own shape ($\triangle$ for Player 1, $\square$ for Player 2) in an empty square, making sure to place it with the number $4$ facing up.[^5] ``` +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ | | | | | | /\ | | | | / 4\ | | | | r----r | | | | | | +--------------+--------------+--------------+ | | | | | | | | | | | | | | | | | | | | +--------------+--------------+--------------+ ``` Intuitively, this means that "all" of the colour is present in that square; or in other words, that there is a $4$ out of $4 = 100\%$ probability that the player's symbol ($\triangle$ or $\square$) would be observed to be in that square. If classical moves were the only moves available to the players, there would be almost no difference between this game and classical tic-tac-toe.[^3] ### The Superposition Move It is time to bring in our first quantum move: the *superposition* move. Here, the player places their piece in a superposition of being in "two places at once". Here's how: Instead of placing a single die with a $4$ up like in the classical move, the player now places *two* dice of the same colour, each in an empty square on the board, and both with the number $2$ up. ``` +--------------+--------------+--------------+ | | | | | | | /\ | | | | / 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 $2$ out of $4 = 50\%$ 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 also know that it is definitely not in the other.) However, until it is *observed* as part of the observation phase of the game, the piece will remain in a superposition of being in both squares at once. ### 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, that is for each square containing a die with a $4$ up, remove the die and draw (or place if you use tokens) the corresponding symbol in the square: $\triangle$ for Player 1 and $\square$ for Player 2. 3. For each piece in a superposition (that is, for each colour that is present in two different squares with the number $2$ up), we roll a d6 to see where the piece ends up. This is done by answering a yes/no question: The player points at one of the squares and asks "is this square mine?", and rolling. A result of an even numbers means "yes", and an odd number means "no". If the answer is no, then the piece must be in the other square. 4. When there are no more dice on the board, *decide* the game. If, by the end of this procedure, one player has a three-in-a-row, that player is declared the winner. If *both* players have three-in-a-rows (which, unlike classical tic-tac-toe, is a possibility in this game), then the player with more three-in-a-rows wins. If there are no three-in-a-rows or the players have the same number of three-in-a-rows, the game is undecided. If the game is undecided and there are no more empty squares left, then there are no more legal moves, and the game ends in a tie. If there *are* more empty squares left, then the game continues, and the turn goes to the next player after the initating player. The available moves are the same as before, and when there are no empty squares left, the observation phase is initiated a second time. Squares that have already been filled in have been *observed*, and the players can not interact with them any further. Let's look at an example. Consider the following board: ``` +--------------+--------------+--------------+ | | | | | /\ | /\ | /\ | | / 4\ | / 4\ | / 2\ | | r----r | g----g | b----b | | | | | +--------------+--------------+--------------+ | | | | | /\ | R---R | G---G | | / 2\ | | 4 | | | 4 | | | b----b | R---R | G---G | | | | | +--------------+--------------+--------------+ | | | | | B---B | /\ | Y---Y | | | 4 | | / 4\ | | 4 | | | B---B | y----y | Y---Y | | | | | +--------------+--------------+--------------+ ``` All but one piece is in a classical *state*,[^4] namely the piece represented by the two blue d4 dice, which is in a superposition state of being in the left square and in the top right square. We follow the procedure outlined above, and start by replacing all classical pieces with their respective symbol. ``` +--------------+--------------+--------------+ | | | | | /\ | /\ | /\ | | / \ | / \ | / 2\ | | +----+ | +----+ | b----b | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / 2\ | | | | | | | | b----b | +---+ | +---+ | | | | | +--------------+--------------+--------------+ | | | | | +---+ | /\ | +---+ | | | | | / \ | | | | | +---+ | +----+ | +---+ | | | | | +--------------+--------------+--------------+ ``` Next, we phrase a yes/no-question. Player 1 points to the top-right square and asks, "is this square mine?" Then they roll. They get a $2$: This is an even number, and so the answer is *yes*. We draw a $\triangle$ in the square, and remove both dice from the board. ``` +--------------+--------------+--------------+ | | | | | /\ | /\ | /\ | | / \ | / \ | / \ | | +----+ | +----+ | +----+ | | | | | +--------------+--------------+--------------+ | | | | | | +---+ | +---+ | | | | | | | | | | | +---+ | +---+ | | | | | +--------------+--------------+--------------+ | | | | | +---+ | /\ | +---+ | | | | | / \ | | | | | +---+ | +----+ | +---+ | | | | | +--------------+--------------+--------------+ ``` We see that $X$ has a three-in-a-row: Player 1 is the winner! If, instead, they had rolled a $3$, the answer would have been *no*: The piece is *not* in the top-right square, and so it must be in the left square. ``` +--------------+--------------+--------------+ | | | | | /\ | /\ | | | / \ | / \ | | | +----+ | +----+ | | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / \ | | | | | | | | +----+ | +---+ | +---+ | | | | | +--------------+--------------+--------------+ | | | | | +---+ | /\ | +---+ | | | | | / \ | | | | | +---+ | +----+ | +---+ | | | | | +--------------+--------------+--------------+ ``` There are no more dice on the board, so the observation phase concludes with the game undecided, *and* there is still an empty square left. The game continues! Now, the turn goes to the opposite player of the initating player (as indicated by the observation token). Whoever they are, it is their lucky day, for we see that now, whoever captures the final square will be declared the winner! ``` +--------------+--------------+--------------+ | | | | | /\ | /\ | R---R | | / \ | / \ | | 4 | | | +----+ | +----+ | R---R | | | | | +--------------+--------------+--------------+ | | | | | /\ | +---+ | +---+ | | / \ | | | | | | | | +----+ | +---+ | +---+ | | | | | +--------------+--------------+--------------+ | | | | | +---+ | /\ | +---+ | | | | | / \ | | | | | +---+ | +----+ | +---+ | | | | | +--------------+--------------+--------------+ ``` ### Balance and Strategy Like in the classical game, this game is all about blocking every opportunity the opponent has at winning. In fact, blocking is in a sense *easier* than before, since one can block two squares at once using the superposition move! However, using the superposition move comes with a certain risk: As seen in the example above, if you don't have a guaranteed three-in-a-row before the observation phase, and the dice land the wrong way, you risk the opponent reclaiming victory. You may have noticed that whoever is first to play after the observation phase has a lot of power. It turns out that the first player can ensure being first after an observation phase by playing in such a way that the second player has no choice but to place last. If they also succeed in blocking Player 2's chances at a three-in-a-row, they may be able to steal a victory—or at least guarantee a tie. So, while the outcome may not be pre-determined (unlike classical tic-tac-toe), this level still has a definite first-player advantage. Therefore, to ensure balanced play, the game should be played twice (or an even number of times), with the players taking turns to go first. ### Levelling Up In light of the above, it is natural to wonder, does *all* levels of TiqTaqToe have a first-player advantage? Well, though we haven't been able to rule it out, already at the next level the first-player advantage appears to mostly vanish, making for a much more balanced and exciting game. So once you have played a few rounds of TiqTaqToe with Superpositions and have gotten comfortable with the flow of the game, it is time to graduate to **Level 2: [[TiqTaqToe with Entanglement]]!** [^1]: 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) [^2]: This version of the game is also playable on [tiqtaqtoe.com](https://tiqtaqtoe.com) under the mode "Minimal Quantumness". [^3]: I say "almost", because you could still get a tie from both players having three-in-a-rows, due to the fact that the game would still only be decided once the board is full; such ties are not possible in standard tic-tac-toe. [^4]: This game uses many potentially unfamiliar terms; see [[TiqTaqToe Player Reference]] for a glossary. [^5]:The players have free choice of which colour to use on each turn, but the following diagrams always have the players use the colours in the following order: Red, green, blue, yellow, purple. In the diagrams, $\triangle$ (d4 dice, lowercase letters) always goes first.