## Level 5 — Master: Fully Quantum TiqTaqToe
We have reached the end of the road, and it is time for you to graduate to TiqTaqToe Master. As a master, you will finally leave the classical world behind, embracing our quantum universe as you expertly manipulate quantum states, bases and observations to bend the pieces towards your goal: Observing three of your pieces aligned in a row.
This is because the main difference to previous levels is that we no longer make a distinction between an observed symbol and a classical state: No more drawing $X$s and $O$s into the squares on the board (other than as a tool during the observation phase). Instead, we simply place a die there with the number $4$ up, as if a classical move had been made.
This means that any state can be interacted with, even after the observation phase, as long as the "max two dice per square" and "no self-entanglement" rules are respected.
In addition, players may now switch between the standard and the skew basis *on every turn*, making squares that are non-interactable in one basis, suddenly interactable again!
Finally, on this level *there is no notion of a tie*: The game *only* ends when a winner has been declared. Should the board be filled up even after the observation phase, with still no winner declared, then the board will simply be observed in alternating bases until one player wins.
### Handing Out the Pieces
As on the previous level, each player gets four d4 dice of each colour: Two with blank $2$s, one with a red $2$, and one with a blue $2$. Take out the observation token, and place it next to the board with the "XO" symbol up, showing that the game begins in the standard basis.
![[IMG_0136.jpeg|400]]
### Switching Basis
On every turn after the first,[^2] before the player makes their move, they may choose to switch the basis, from standard to skew, or from skew back to standard.
When switching basis, first flip the observation token so that the "+-" side faces up. Then make the following exchanges: For any *independent* squares (meaning no entanglement), exchange plus-states for $X$ states, and minus-states for $O$ states.
$\text{Before}$
![[IMG_0138.jpeg|400]]
$\text{After}$
![[IMG_0139.jpeg|400]]
These states, which were previously non-interactable due to the "max two dice per square" rule, may now be interacted with again: Entangled, cancelled, or spun!
Similarly, $X$ states are exchanged for plus-states, and $O$ states are exchanged for minus-states. This is done using dice of the same colour as the die that occupied the square to begin with: A plus-state can be constructed with a blue $2$ and a red $2$—this gives the state "minus half $X$ minus half $O
quot;, and since we can multiply any state by $-1$ without changing it, this is the same as "half $X$ plus half $Oquot;. Similarly, minus-states can be represented using one blank and one red $2$ of same-coloured dice, which gives the state "half $X$ minus half $Oquot; or "half $X$ minus half $Oquot;—which, again, are both equivalent to a minus-state.
$\text{Before}$
![[IMG_0142.jpeg|400]]
$\text{After}$
![[IMG_0144.jpeg|400]]
And so we see the opposite also holds: States that were interactable in the standard basis, are now *non*-interactable!
Superposition states are switched similarly, except now the states should be *half* plus or minus states, which should be represented using red and blue $1$s for half plus-states, and red and blank $1$s for half minus-states. In the current implementation of the game, since we don't have more $1$s to spare, these are represented using red and blue $3$s.
$\text{Before}$
![[IMG_0146.jpeg|400]]
$\text{After}$
![[IMG_0148.jpeg|400]]
Cancelled squares remain cancelled, and entangled squares remain entangled.
$\text{Before}$
![[IMG_0150.jpeg|400]]
$\text{After}$
![[IMG_0151.jpeg|400]]
Finally, the half of a half-entangled or half-cancelled superposition that has been interacted with does not change. This is because in either basis, they will be a non-trivial superposition over both symbols and squares. However, the untouched half becomes a half $+$/half $-$ state, as above.
$\text{Before}$
![[IMG_0154.jpeg|400]]
$\text{After}$
![[IMG_0156.jpeg|400]]
### The Observation Phase
The observation phase proceeds similarly to before, with a few changes. First, give the observation token to the player that initiated the phase. Next, if the board is in the skew basis, switch it back to the standard basis. Then, the observation phase proceeds exactly like in [[TiqTaqToe with Complementarity]], until all pieces have been observed and it has been decided whether there is a winner, or if the game continues.
Let us observe the following board.
![[IMG_0157.jpeg|400]]
We see from the observation token that the board is currently displayed in the skew basis, so we start by switching it back to the standard basis.
![[IMG_0158.jpeg|400]]
Next, we roll to figure out which squares are empty. We remove the dice from any cancelled squares,
![[IMG_0159.jpeg|400]]
and roll for the lower-left square, since it has a $50\%$ chance of being empty. Player 1 points to it and asks, "is this square mine?", and rolls a $1$; it is *not*.
![[IMG_0160.jpeg|400]]
Now, the remaining squares will all contain a piece after observation. Next, Player 2, who initiated the observation phase, chooses a basis to observe in. Here they see that they have a chance to capture the middle horizontal line, and so they choose the *skew* basis.
At this point, the observation proceeds as normal, including writing symbols in the measured squares, in order to keep track of which squares have been resolved and which ones remain to be rolled. First, flip the observation token to the skew basis, and replace the plus and minus states with their respective symbols.[^4]
![[IMG_0161.jpeg|400]]
Let us resolve the entangled pieces next. Player 2 points to the top-left square and asks, "is this square mine?", and roll a $6$.
![[IMG_0163.jpeg|400]]
Two independent rolls remain. Player 1 rolls for the top square and rolls a $6$,
![[IMG_0164.jpeg|400]]
and Player 2 rolls a $2$ for the bottom-right square.
![[IMG_0165.jpeg|400]]
A very even board indeed!
But a lot could still happen, because, since a winner was not declared, the final step of the ovservation phase is to replace all of the symbols with pieces in classical states. Then, the turn goes to Player 1!
![[IMG_0166.jpeg|400]]
What should Player 1 do? Should they make a superposition move in the two empty squares, and thus go straight to another observation phase? If so, they will win if they observe in the skew basis (which is where we currently are), and their piece lands in the middle; otherwise, Player 2 will take the victory. 50/50 in other words; can they do better?
Maybe they will prolong the game by cancelling one of Player 2's pieces? The top left square is involved in two possible winning moves for $O$, so cancelling this while attempting to capture the middle might be a good move.
Or they might upend the whole playing field by going straight to the observation phase, via a superposition move as before, but instead observe in the *standard* basis. If so, every single piece currently on the board would have to be independently re-rolled. The outcome of such a move is anyone's guess!
What would you do?
### Balance and Strategy
In terms of possible moves, this level is the same as the previous, so the same considerations still apply: Do you race to the end to ensure *you* get to choose the observation basis? Or do you aim to gain an advantage in both bases simultaneously, with a mere *hope* that the choice of basis falls to you?
The ability to switch bases back and forth can be of great aid in the latter effort, and knowing that the opponent may switch the basis any time means that moves such as the spinning move—which were previously safe from interaction thanks to the "max two dice" rule—are now open to attacks, such as cancelling.
Here, then, the entangling move makes something of an unexpected return as a strategic move, since entangled pieces are unaffected by the basis change. In this sense, entanglement *protects* the pieces from further attacks![^1]
What other tactics can be applied at this level remains largely unexplored, and so we leave the next, great strategy up to *you* to discover!
##### Regarding Neverending Games
Unlike previous levels, at this level it is possible to play in such a way that the game never ends: If Player 1 starts by playing a classical move, then the players can take turns continuously cancelling each other every turn by switching basis, so that the remaining piece becomes interactable again, and cancelling it, until the board is filled with dice without a second piece ever having been placed. Then, after the observation phase, there will again be a single piece on the board in a classical state, which can again be cancelled!
I think players will agree that there are more fun ways to play the game than this. Still, it is possible that this level contains undiscovered corner cases in which the optimal move for both players is to play in such a way that the game never ends![^3]
### Where To Next?
We have now reached the top of the mountain, and so all that remains for me is to congratulate you on your new title as *TiqTaqToe Master!* I hope you have found the learning experience as entertaining and stimulating as I have found developing and playtesting this ruleset over the past year.
TiqTaqToe is but one of many quantum games. Among these, we usually distinguish between "quantum-inspired" and "true quantum" games, where the former mainly aims to *illustrate* quantum concepts, with Schrödinger's Cat scenarios and the like, while the latter builds games using the real laws of quantum physics. This means, among other things, that the game could easily be implemented on a quantum computer with the same ruleset. TiqTaqToe is one of few such games—but it is not the only one!
In particular, TiqTaqToe's origins were heavily influenced by its more mature and complex bigger brother, [Quantum Chess](https://quantumrealmgames.com/). If you enjoyed TiqTaqToe, and feel ready to take a plunge into the *full* set of laws governing our quantum nature—including complex phases that sit "in between" plus and minus, and arbitrarily large superpositions—then this may be the natural next step: Quantum Chess is available [online](https://quantumrealmgames.com/play/) and on [Steam](https://store.steampowered.com/app/453870/Quantum_Chess/) for all platforms, and the game supports local and online multiplayer, as well as single-player against AI. There are even puzzles!
Meanwhile, I have repeatedly found that the best way to gain a deeper understanding of TiqTaqToe is to introduce it to new players. So pack your dice in your pocket, go out in the world, and find new and curious people to play against!
After a game or two, maybe they too will be inspired to walk **[[The Road to TiqTaqToe Mastery]]**.
[^1]: Here's a question to test your understanding: Is an entangled state (meaning two blank $2