## Level 1 – Beginner: TiqTaqToe with 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 $X$s and $O$s on a 3 by 3 grid, and the first player to get three symbols 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 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 d4 dice. Each player has dice of five colours, and there are four dice of each colour. For this version though, only two dice are needed per colour, so start by putting away the dice with blue and red $4$s and $2$s. Each player should then have ten d4 dice. ![[IMG_9733 1.jpeg|400]] Each player also gets one d6 die, to be used in the observation phase, and the *observation token* should be placed between the players, with the "XO" side facing up. *(bilde)* ### Classical Moves For each turn, the player places dice of a colour that has not already been placed. When making a classical move, the player places *one* such die in an empty square, making sure to place it with the number $4$ facing up. ![[IMG_9736.jpeg|400]] 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 ($X$ or $O$) will be found to be in that square in the end. 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] ### Our First Quantum Move But another move is available: 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, both with the number $2$ up. ![[IMG_9737.jpeg|400]] This shows 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 will be found to be in one or the other square in the end. However, until *observed*, 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. Place the observation token next to the initating player with the "XO" side facing up. 2. For each classical piece, that is for each square containing a die with a $4$ up, remove the die and draw an $X$ (resp. $O$) in its place. 3. For each piece in a superposition, that is for each colour that is present in two different squares with numbers $2$ up, we roll to see where the piece ends up. This is done by answering a yes/no question, like "is the piece in the *this* square?", and rolling the d6: Even numbers mean "yes", odd numbers mean "no". If the answer was no, then the piece is 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 is a possibility in this game not found in the classical 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, so the player that does *not* have the observation token. Place the observation token next to the board again (still with the "XO" side up). The available moves are the same as before, and when there are no empty squares left, the observation phase is initiated again. However, squares that have already been filled in have already been *observed*, and the players can not interact with them any further. Let's look at an example. Consider the following board: ![[IMG_9740.jpeg|400]] All but one *piece* is in a classical *state*,[^4] namely the piece represented by a green die, which is in a superposition state of being in the leftmost square and in the top-right square. We follow the procedure outlined above, and start by replacing all classical pieces with their respective symbol. Here, dice with white numbering belong to Player 1 ($X$), and dice with gold numbering belong to Player 2 ($O$). ![[IMG_9741.jpeg|400]] Next, we phrase a yes/no-question. Player 1 points to the top-right square and asks, "is this square mine?" Then they roll. ![[IMG_9742.jpeg|400]] They get a $2$: This is an even number, and so the answer is *yes*. We draw an $X$ in the square, and remove both dice from the board. ![[IMG_9743.jpeg|400]] 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 leftmost square. ![[IMG_9744.jpeg|400]] There are no more dice on the board, so the observation phase concludes with no winner declared, *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! ![[IMG_9745.jpeg|400]] ### Balance and Strategy Like in the classical game, this game is all about blocking every opportunity the opponent has at winning. Additionally, blocking the opponent 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 has a definite first-player advantage. Therefore, to ensure balanced play, the game should be played twice (or an even number of times), with each player taking turns going first. ### Levelling Up %% In light of the optimal strategy, it is natural to wonder: Will an optimal strategy *always* exist? Well, the answer is, no one knows! In fact, by introducing a second quantum move, we already get a game for which it remains *unknown* whether a optimal strategy exists! %% In light of the above, it is natural to wonder, will there *always* be a first-player advantage? The answer is: We don't know! But already at the next level, the first-player advantage appears to vanish—or at least be significantly weakened, making for a much more balanced and exciting game. So once you have played a few rounds of [[TiqTaqToe with Superpositions]], and feel comfortable with this "beginner"-level of the game, it is time to graduate to **Level 2: [[TiqTaqToe with Entanglement]]!** [^1]: For an example of a solved game that *is* still interesting, because the optimal strategy is too complicated for a human to implement, see [Quartic](). [^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. [^4]: This game uses many potentially unfamiliar terms; see [[TiqTaqToe Player Reference]] for a glossary.