I looked for a great Tic-Tac-Toe variant, and managed to stumble across it, thanks to the famous article on Math with Bad Drawings.

Mr. Orlin's description of the rules of the game have no obvious room for improvement, so instead of taking the time to write a worse description, this post will build on his foundation to describe terms, openings, materials, and other items that I've found useful as a player.

The game proper is referred to long-form as "Ultimate Tic-Tac-Toe", or as merely "Ultimate", but we've taken to calling it "ULT", a short form which distinguishes from other games (like the frisbee tossing game).

A square that keeps the play on the same board (for instance, the center square on the center board) is called a 'home square'.

The ability to place a piece on any board is referred to as a 'free move', and a square that allows your opponent to perform this move can be called a 'free move square'.

There are several means of notation that I have discovered or thought of:
1) Algebraic: Like algebraic notation on a chess board, with letters corresponding to columns and numbers to rows. The bottom-left square is a1, the center square is e5, and the top-right is i9.
2) Compass: Uses directions like "NW", "SE", or "C". The board picked is first, then the square on the board. The bottom-left square is SW/SW, the center is C/C, and the top-right is NE/NE.
3) Numeric : Each square has a number. The first number is the board you are playing on, the second is the square you are moving on. The orientation may vary depending on whether you are using a telephone-style keypad or a computer/calculator-style keypad; the latter is my default. The bottom-left square is 11, the center is 55, and the top-right is 99.
4) Relative : Like the compass system, but uses arrow symbols instead of letters. This can help in-person play, because the arrows will be correct from each player's perspective. Bottom-left would be ↙↙, center would be ••, and top-right would be ↗↗.

I prefer the numeric system in principle, but my games tend to be written in algebraic notation, because I was familiar with it from chess and didn't think of a better way to do it until after I had a large body of work in the former.

For denoting a move that wins the game, or trivially forces a win, the convention is to use # following your move, the same as in chess checkmate.

This is not standard notation, but a move that captures a board may be denoted by putting '[X]' (or perhaps '[O]') at the end of it. Likewise, a move that causes a free move may be noted by "FM". If another symbol is chosen, I would recommend an asterisk [*] or perhaps an at symbol [@].

Yu can refer to a move that captures a board as being '3r', and a move that wins the game as being '3br'.

Playing Materials

In online play, I use a spreadsheet and exchange moves with my opponent via IRC or a similar protocol. I developed a custom spreadsheet some time back, which helps keep track of free move squares, allows convenient text input of moves, and can display captured boards. Click here to download it for yourself. The first page also includes match details, useful for a series of games.

There are several computer implementations that manage the board and can help arrange online games. The most popular one used to be Bennett Zhang's Heroku app, but it seems to be offline, as of 9/1/23. Arkadiusz Nowaczyński's uttt.ai includes useful analysis options, as well as probably the strongest AI that exists, which plays as well as the strongest chess computers play chess. Ofek Gila's program is older, but also features a strong AI.

For in-person play, you can either use pen-and-paper, or manufacture a board and pieces. If you don't mind using printer ink, you can expedite the process of board-making a little bit by printing out "Game Gal's" boards.

Misc. Norms, Tips, et al.

A player familiar with the game is able to play at a 'classical' level in 25-45 minutes. Two games in 60-70 minutes is a decent pace with sufficient time for thought. If a game goes on longer than an hour, someone is taking up too much time. A game shouldn't go on as long as an hour and a half.

As in regular Tic-Tac-Toe, the first mover (by convention, X) has an advantage. Like Chess, the game is not proven to be a draw with perfect play, but it likely is. In a fair match, each player should have an equal number of games as first and second mover.

Mr. Orlin mentions a variation of the game where 'tie' boards are considered won for both players (whereas the norm is for those boards to be considered inert). We played this several times as "Cutthroat" (or "CUT"). Most games seem to be unaffected, but there are positions where this makes all the difference. I would stick with 'inert' boards for ties, on the grounds that it is mainstream and more intuitive. If you do play CUT, the norm we figured is that in the case of simultaneous opposing 3brs, the game is drawn.

Another variation is known where the play is the same, but the objective of the game is to win 5 boards. This does affect the outcome of the game, for similar reasons as why a popular vote can result in different outcomes from the electoral college. While it is ULT-like, I would not call this game "Ultimate" or "ULT". The game seems to have been named "Tic-Tac-Ku", and I would stick with that, for the purpose of distinguishing the two.

General Strategy

Several articles could be written on this subject, and perhaps will be in the future, but here I will try to give a brief overview.

This game is a new frontier in many ways, and there's less known about play and principles of play than I would like. We are all students of the game, and it is best to avoid being dogmatic about an approach. A type of move that works in one case may turn out to be a disaster in another. Any advice given can only be very general, subject to circumstances.

From our earlier study of tic-tac-toe, we know that the center square is mathematically the strongest square on a tic-tac-toe board. Therefore, we understand that the center board is the strongest board in ULT. The difference is that while the center square is easily defended against in a tic-tac-toe game, good positioning on the center board can provide an enduring advantage that is difficult to offset.

Whenever you move on a square, you are giving your opponent compensating value in the board that you're directing them to. A major principle of play is to get gain in boards you care about while trying to make sure your opponent only gets advantages in boards that you don't. This is complicated by the possibility of your moves being 'wasted' by overplaying on a board that you already have a winning advantage in, or by being forced to play in boards that will lead to a good gain for your opponent.

As boards are won and the game gets to the midgame or endgame, there will be many times when you don't want to move on a certain board, because the only squares you can move on there will give your opponent the chance to make a winning play on another board. From my experience, these positions are seldom clear in advance, and can take up calculation time to uncover.

Never underestimate the power of a free move. Many games have been won by someone being forced to move on a free move square. Don't give up free moves for free.

On the other hand, don't underestimate the power of a good major board position, even if securing it gives up a free move. If your opponent has a 3br threat, it is often worth quenching it, even if that gives them a free move somewhere else. A 3br threat, once it exists, will hang over your head for the rest of the game, because it means that any free move square you land on ends the game. Likewise, you want to have a good 3br threat if you can get one. It's a move that pays you interest.

If you can win or draw on merely three boards - the center, and two opposing corners - that is sufficient to not lose the game, even if your opponent wins every other board. (This is why the outcome of Tic-Tac-Ku is so different from regular ULT.)

Because we can't see everything, it is often a good idea to 'hedge your bets' and not commit to trying to win a certain line of boards (at the expense of the others) too early. A board that seems to have a bad position one turn may turn out to be useful just a few minutes later.

Once boards start being won, it is harder to win others without paying an extra cost. The sheer mass of your opponent's already-won boards, even if they are not particularly good boards, can make all the difference when you're both being forced into tight spaces.

Openings

Because the board is symmetrical, there are fewer openings than there are squares:
Center-Center (e5, 55) - "The Main Line"
Center-Corner (d4/51, d6/57, etc.) - "The Corner Attack"
Center-Edge (e4/52, e6/58, etc.)
Corner-Home Corner (a1/11, a9/77, etc.)
Corner-Alternate Corner (a3/17)
Corner-Opposing Corner (c3/19, etc)
Corner-Center (a2/15, etc.)
Corner-Aligned Edge (a2/12)
Corner-Unaligned Edge (b3/18)
... and a plethora of edge board openings too tedious to mention in detail.
UTTT.ai analysis scores are shown after an opening's name.

Center-Center (e5, 55) - The Center Game - "The Main Line" - +11.81
This is considered the most solid opening. It gives you a decent (though not insurmountable) advantage on the center board, and your opponent's move will give you the option of playing the first move on a second board, probably a corner board. Strong AIs strongly prefer this move.

Center-Corner (d4/51, d6/57, etc.) - The Corner Game - "The Corner Attack" - +11.07
When it comes to alternate openings (besides e5), this was my initial unstudied preference. This move seems to give a greater edge on the center board than e5 does, because of the lack of a piece to oppose your own. I second-guessed this opening for a while, but computer analysis has revealed that it is, in fact, a top-2 opening, as I originally imagined.

Center-Edge (e4/52, e6/58, etc.) - +8.16
The center-edge opening places an uncontested piece on the center board and gives O the lead move in an edge board. The edge square is weak in the center board, because it has only two lines, one of which runs through the center square - which is also weaker than normal because it allows the opponent to immediately reply. I'd suspect I'm not meeting my potential here, but it is likely playable, since the edge board compensation your opponent gets is slim.

Corner-Home Corner (a1/11, a9/77, etc.) - +6.47
Places a contested piece on a corner board, giving X and O near-equality on this board and allowing them to pick the next board, but keeping the first move on that board. This move is not obviously bad and seems, at first glance, not worse than the following options.

Corner-Alternate Corner (a3/17)
Places an uncontested piece on a corner board, allowing O the lead move in another corner. This move completely ignores the center board, leaving that board's status the 'elephant in the room'. This is perhaps one of the purest openings in terms of allowing the player to measure the strength of the first-move advantage, as X and O's positions at this point seem rather equal otherwise.

Corner-Opposing Corner (c3/19, etc) - +6.12
Places an uncontested piece on a corner board, allowing O the lead move in the opposing corner. This move is similar to the previous one, but has the characteristic of weakening your chances for 3br from your initial board, due to giving compensation to your opponent at the other end of the line. I am unsure of it on those grounds.

Corner-Unaligned Edge (b3/18)
Places an uncontested piece on a corner board, allowing O the first move on an edge board that does not overlap with the corner board's rows. It seems plausible that X maintains a slight advantage here and that the move is not outright 'bad'.

Corner-Aligned Edge (a2/12)
Places an uncontested piece on a corner board, allowing O the first move on an edge board that overlaps with the corner board's rows. My intuitive guess is that this position is a little worse than the previous one, although it is hard to measure or project without an in-depth study.

Corner-Center (a2/15, etc.)
Places an uncontested piece on a corner board and gives O the lead move in the center board, which gives them similar prerogatives to the main openings. This move seems to put O in the driver's seat, allowing X to defend with a position stronger than what O could have had. Philosophically I doubt that it is as viable an opening as the others, but it might be a good psychological ploy, or a good move for someone who is used to playing as the second player and would like to reuse as much of their knowledge as possible.

Edge Board Openings
There is a wide variety of possible openings here, given that there's not only a home-square edge, but also corners or edges that do not overlap with the first board's rows, versus ones that do, but they seem to have one thing in common: they're all bad. Any move on an edge board is likely to either result in a seemingly equal position (not clearly accomplishing anything for X), or to even give O the advantage. This might be viable for a handicap game or a pie-rule tiebreaker, but it should be ruled out as a serious move between comparable players.

Conclusion

Check back periodically to see if there have been revisions to this, as there likely will be in the future. In the meantime, be on the lookout for a fourth article; there is still more to say about this game, and also about some close variants of it.