## 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 and modes of observation to bend the state of the board towards your goal: Observing three of your own pieces aligned in a row.
The main change from previous levels is that now, we will no longer make a distinction between an observed symbol and a classical state: No more placing $\triangle$ and $\square$ into the squares on the board; instead, we will simply place a die there with the number $4$ up, as if a classical move had been made.
The result is that any square can be interacted with even after the observation phase, as long as the "max two dice per square" and the "no self-entanglement" rules are respected.
Additionally, players may now switch between the standard view of the board, and a *rotated view*. As for the rotated mode of observation, this switches the roles of classical states and *plus*/*minus* states, allowing players to interact with *plus* and *minus* states as if they were classical!
%%As an optional rule, you may now also *do away with the notion of a tie*. With this rule, the game *only* ends when a winner has been declared. Should there be no empty squares left after an observation phase with no winner declared, then the board is simply observed under alternating modes (standard, rotated, standard, and so on) until a winner emerges.%%
### Distributing the Pieces
Distribute the dice as on Level 4 ([[TiqTaqToe with Complementarity]]), except now place one d6 per colour for Player 1 and one d4 per colour for Player 2 in the bank, so that the bank now mirrors exactly the dice handed out to the players.
```
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
Bank
/\ /\ /\ /\ /\ 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
```
Then take out the *view token*, and place it next to the board with the $\triangle$/$\square$ symbol up.
### The Rotated View
On their turn, before of after they make their move, the player may choose to switch from the standard to the rotated view or from the rotated view back to the standard view. However, **each player may only switch the view once between each observation phase**.
To switch to the rotated view, first flip the observation token so that the $+$/$-$ side is facing up. Then go through the squares one by one, from the top left to the bottom right, and decide whether anything changes (being careful not to lose track of which ones you have updated and which ones remain to be changed).
Make the following exchanges: *Plus* states turn into classical $\triangle$ states and *minus* states turn into classical $\square$ states; $\triangle$ states turn into *plus* states (using the player's own dice to represent the state), and $\square$ states turn into *minus* states; finally, superposition states turn into *plus-superposition* and *minus-superposition* states, which are as plus and minus states but with the numbers $1$ in place of the $2$s. Entangled and cancelled squares do not change, but for half-entangled and half-cancelled states, make the uninteracted half of the state a *half-plus* or *half-minus* state, like for superposition states. For a visual overview of the changes, see [[TiqTaqToe Player Reference]].
As an example, consider the following board:
```
+--------------+--------------+--------------+
| | | |
| /\ | R---R | R---R |
| / 4\ | | 2 | | | 2 | |
| r----r | R---R | R---R |
| | | |
+--------------+--------------+--------------+
| G---G | B---B | /\ |
| /\ | 2 | | /\ |-2 | | /\ /-2\ |
| / 2\ G---G | / 2\ B---B | / 2\ B----B|
| g----g | b----b | b----b |
| | | |
+--------------+--------------+--------------+
| G---G | | |
| /\ | 2 | | | |
| / 2\ G---G | | |
| g----g | | |
| | | |
+--------------+--------------+--------------+
```
We have one classical state (top left), one superposition state (top and top right), two entangled squares (left and bottom left), one cancelled square (right), and a *minus* state (middle), in addition to two empty squares.
It is Player 1's turn. In the standard view, only the superposition state is open to be interacted with, so the choices are to either place make a classical or superposition move in the open squares, or to perform a half-entangling or half-cancelling move on Player 2's red (R) superposition state.
However, by *rotating* the view, another option opens up.
Going square by square (top left to bottom right):
1. A classical $\triangle$ state; turn into a *plus* state by moving the d4 to a $2$, and placing a d6 of the same colour (from Player 1's own dice pool) with a $2$ facing up.
2. A superposition $\square$ state; turn into a *half-plus* state by moving the d6 to a $1$, and placing a d4 of the same colour (from Player 2's own dice pool) with a $1$ facing up.
3. The other half of the superposition state; repeat the previous step.
4. This square is entangled with the bottom left square (square 7); do nothing.
5. This square contains a *minus* state; turn into a classical $\square$ state by removing the d4 and moving the d6 to a $4$.
6. This square is cancelled; do nothing.
7. This square is entangled with the left square (square 4); do nothing.
8. Empty
9. Empty
The result should look like this:
```
+--------------+--------------+--------------+
| r---r | R---R | R---R |
| /\ | 2 | | /\ |-1 | | /\ |-1 | |
| / 2\ r---r | / 1\ R---R | / 1\ R---R |
| r----r | R----R | R----R |
| | | |
+--------------+--------------+--------------+
| G---G | | /\ |
| /\ | 2 | | B---B | /\ /-2\ |
| / 2\ G---G | | 4 | | / 2\ B----B|
| g----g | B---B | b----b |
| | | |
+--------------+--------------+--------------+
| G---G | | |
| /\ | 2 | | | |
| / 2\ G---G | | |
| g----g | | |
| | | |
+--------------+--------------+--------------+
```
Suddenly, the middle is now open for interaction, letting Player 1 entangle with it or cancel it at their leisure. Meanwhile the squares along the top row, which were all previously interactable, are now blocked from interaction, at least until the view is switched back from rotated to standard.
What would *you* do next?
#### View Change as a Defence Mechanism
In the example above, Player 1 switched to the rotated view in order to open a line of attack on the middle square. However, since each player only gets to switch the view *once* between each observation phase, switching the view can also be used to defend a piece from being interacted with.
For example, Player 1 may start by placing a classical piece in the middle square
```
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | | |
| | /\ | |
| | / 4\ | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
```
and then immediately switch the board to the rotated view.
```
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
| | r---r | |
| | /\ | 2 | | |
| | / 2\ r---r | |
| | r----r | |
| | | |
+--------------+--------------+--------------+
| | | |
| | | |
| | | |
| | | |
| | | |
+--------------+--------------+--------------+
```
Now, if Player 2 wants to interact with the square (cancel or entangle with it), they will first have to switch back to the standard view. However, they only get to do this *once* before the observation phase. Better then to wait and see how the board develops, or to go full steam ahead and spend that resource to get at the middle square right away while they still have free choice of squares to use?
Player 1, meanwhile, have of course spent their one view change to defend their middle position; time will tell whether this turned out to be a tactical move or not.
### The Observation Phase
The observation phase proceeds as before, except for a few changes. First, if the board is in the rotated view, then switch it back to the standard view. Then, proceed with the observation phase exactly as in [[TiqTaqToe with Complementarity]], until all pieces have been observed and it has been decided whether there is a winner. Finally, if no winner is declared, replace all the observed tokens with classical $\triangle$ and $\square$ states, using dice from the respective player's dice pool. Then game continues, with the turn going to the next player.
Let us observe the following board.
```
+--------------+--------------+--------------+
| B---B | G---G | B---B |
| /\ | 2 | | /\ |-2 | | /\ | 2 | |
| / 1\ B---B | / 2\ G---G | / 1\ B---B |
| b----b | G----G | b----b |
| | | |
+--------------+--------------+--------------+
| Y---Y | /\ | | ___
| /\ | 2 | | /\ /-2\ | R---R | // \
| / 2\ Y---Y | / 2\ R----R| | 4 | | || +/- |
| y----y | r----r | R---R | \\___/
| | | |
+--------------+--------------+--------------+
| b---b | Y---Y | g---g |
| /\ | 1 | | /\ | 2 | | /\ | 2 | |
| / 1\ b---b | / 2\ Y---Y | / 2\ g---g |
| b----b | y----y | g----g |
| | | |
+--------------+--------------+--------------+
```
**Step 1: Set the view to standard**
The view token shows $+/-$, telling us that we are currently seeing the board under the rotated view, so we start by switching it back to the standard view. Following the procedure as explained above, we end up with the following board:
```
+--------------+--------------+--------------+
| B---B | | B---B |
| /\ | 2 | | G---G | /\ | 2 | |
| / 1\ B---B | | 4 | | / 1\ B---B |
| b----b | G---G | b----b |
| | | |
+--------------+--------------+--------------+
| Y---Y | /\ | R---R | ___
| /\ | 2 | | /\ /-2\ | /\ |-2 | | // \
| / 2\ Y---Y | / 2\ R----R| / 2\ R---R | || △/□ |
| y----y | r----r | R----R | \\___/
| | | |
+--------------+--------------+--------------+
| | Y---Y | |
| /\ | /\ | 2 | | /\ |
| / 2\ | / 2\ Y---Y | / 4\ |
| b----b | y----y | g----g |
| | | |
+--------------+--------------+--------------+
```
**Step 2: Identify empty squares**
We start by removing the dice from any cancelled squares,
```
+--------------+--------------+--------------+
| B---B | | B---B |
| /\ | 2 | | G---G | /\ | 2 | |
| / 1\ B---B | | 4 | | / 1\ B---B |
| b----b | G---G | b----b |
| | | |
+--------------+--------------+--------------+
| Y---Y | | R---R | ___
| /\ | 2 | | | /\ |-2 | | // \
| / 2\ Y---Y | | / 2\ R---R | || △/□ |
| y----y | | R----R | \\___/
| | | |
+--------------+--------------+--------------+
| | Y---Y | |
| /\ | /\ | 2 | | /\ |
| / 2\ | / 2\ Y---Y | / 4\ |
| b----b | y----y | g----g |
| | | |
+--------------+--------------+--------------+
```
then roll for the lower-left square, since that square has a $50\%$ chance of being empty. Player 1 points to it and asks, "is this square mine?", and rolls a $1$; the answer is no, and the square is empty. The top left and top right square update accordingly:
```
+--------------+--------------+--------------+
| B---B | | B---B |
| /\ | 2 | | G---G | /\ | 2 | |
| / 2\ B---B | | 4 | | / 2\ B---B |
| b----b | G---G | b----b |
| | | |
+--------------+--------------+--------------+
| Y---Y | | R---R | ___
| /\ | 2 | | | /\ |-2 | | // \
| / 2\ Y---Y | | / 2\ R---R | || △/□ |
| y----y | | R----R | \\___/
| | | |
+--------------+--------------+--------------+
| | Y---Y | |
| | /\ | 2 | | /\ |
| | / 2\ Y---Y | / 4\ |
| | y----y | g----g |
| | | |
+--------------+--------------+--------------+
```
At this point, we know that the remaining non-empty squares will all contain a piece after observation.
**Step 3: Choose a mode of observation**
They player chooses the rotated mode, then proceed as described in [[TiqTaqToe with Complementarity]]: $+$ and $-$ states are replaced by $\triangle$ and $\square$ tokens, while classical as well as entangled states that are rolled for.
After all the rolls are made, the board looks as follows:
```
+--------------+--------------+--------------+
| | | |
| /\ | /\ | +---+ |
| / \ | / \ | | | |
| +----+ | +----+ | +---+ |
| | | |
+--------------+--------------+--------------+
| | | |
| /\ | | +---+ |
| / \ | | | | |
| +----+ | | +---+ |
| | | |
+--------------+--------------+--------------+
| | | |
| | +---+ | /\ |
| | | | | / \ |
| | +---+ | +----+ |
| | | |
+--------------+--------------+--------------+
```
For Player 2, the situation is looking dire. However, a lot could still happen, thanks to—
**Step 4: Place observed pieces in classical states**
Which coloured dice are used for each classical state does not matter, as long as the dice belong to the corresponding player.
```
+--------------+--------------+--------------+
| | | |
| /\ | /\ | R---R |
| / 4\ | / 4\ | | 4 | |
| r----r | g----g | R---R |
| | | |
+--------------+--------------+--------------+
| | | | ___
| /\ | | G---G | // \
| / 4\ | | | 4 | | || △/□ |
| b----b | | G---G | \\___/
| | | |
+--------------+--------------+--------------+
| | | |
| | B---B | /\ |
| | | 4 | | / 4\ |
| | B---B | y----y |
| | | |
+--------------+--------------+--------------+
```
The game continues!
If the turn goes to Player 1 next, they can make a move that guarantees them victory. Do you see it?
If on the other hand the turn were to go to Player 2, how could they save themselves? Well, Player 1 has two winning options, and both of them involve the top left square. Maybe Player 2 can do something to sabotage that square? This may not open for a victory, but could at least save the tie.
Alternatively, Player 2 *could* upend the whole board by going straight to the observation phase via a superposition move, and then observe the board in the rotated mode of observation. If so, *every* piece on the board would have to be rolled for. The result such a move would be 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 mode of observation? Or do you aim to gain an advantage in both views simultaneously, with a mere *hope* that the choice of mode falls to you?
The ability to switch views during the game can be of great aid in the latter effort. However, with each player only being allowed to switch once between observation phases, this capability is a scarce resource, not to be spent lightly. What's more, while switching view opens certain states to attack, other states, like the classical and superposition states, suddenly become non-interactable!
Nonetheless, moves that were previously safe against interaction do become open to attack after a view change. It is here that the entangling move makes something of an unexpected strategic return, since entangled pieces are unaffected by view changes. 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 discovering the next, great strategy up to you!
%%##### 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]
[^3]: If you come across such a case, I would be very curious to see it, so shoot me an email at hans (at) heum (dot) me if you're willing to share!
This possibility is also removed by restricting the number of times each player can switch the view during play. We recommend restricting to once or twice per player between observations. (Such a restriction also makes the mechanic more interesting, since now it may actually be a problem that some states become non-interactable when switching, since they can't just as easily switch back.)%%
### 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.
TiqTaqToe is but one of many quantum games. Among these, we often distinguish between "quantum-inspired" and "true quantum" games, where the former aims to merely *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 (and one of very few such *board games*)—but it is not the only one.
In particular, TiqTaqToe's origins were heavily influenced by its 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 many splits of a superposition—then this may be the right next step for you. Quantum Chess is available to play [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!
Who knows, after a few games, maybe they too will decide to walk **[[The Road to TiqTaqToe Mastery]]**!
[^1]: Here's a question to test your understanding: Is an entangled state (meaning two $2