The square variant is a Quarto rule in which four pieces sharing a trait also win when they fill a 2×2 block of adjacent cells, not just a row, column or diagonal. It adds 9 winning shapes to the classic 10, for 19 in total. Many Quarto editions print it as an option, and Quads always plays this way. Our numbers show it makes the game far sharper: a board filled at random has no winning shape only 0.31% of the time with squares, against 1.98% without. In exact endgame solving, draws with seven pieces placed fell from 42.9% to 5.8%.
About us: we make Quads, an independent game built on the same hand-your-opponent-the-piece mechanic as Quarto. It isn't affiliated with Gigamic or the makers of Quarto.
The rule
Classic Quarto is won by placing a piece that completes a row, column or diagonal of four pieces sharing at least one trait. The square variant keeps all of that and adds one more way to win:
- Any 2×2 block of four adjacent cells counts as a winning shape.
- As with lines, the four pieces must share at least one trait: all tall, all dark, all hollow, and so on.
- Everything else is unchanged. You still place the piece your opponent hands you, then hand them one.
The variant appears in the rules of many Quarto editions. Wikipedia describes it as "a variant rule included in many editions". If you play on a physical set, agree before the game whether squares count.
Where the squares are
A 4×4 board has 9 blocks of 2×2: three across times three down. They overlap, so the cells aren't equal. A corner cell is in 1 block, an edge cell in 2 and a centre cell in 4. Add the lines and each centre cell is part of 7 winning shapes, against 4 for every other cell.
Winning shapes through each cell with the square rule
Squares also change the board's symmetry. We searched every rearrangement of the 16 cells: classic rules have 32 that keep every line a line, but only 8 keep every line a line and every square a square. Under classic rules a corner and a centre cell are interchangeable. With squares they're not. Our cell-by-cell analysis has the details.
How much more often someone wins
We filled 1,000,000 boards with the 16 pieces in random order and counted the winning shapes on each:
| Classic (10 lines) | With squares (19 shapes) | |
|---|---|---|
| Random full boards with no winning shape | 1.98% | 0.31% |
| Winning shapes on an average full board | 2.94 | 5.60 |
| Placement on which random play first wins (average) | 11.6 | 10.5 |
Squares almost double the number of winning shapes on a full board, and they cut the chance of a random "dead" board by a factor of about 6. They also bring the first win about one placement earlier.
Endgames with perfect play
Random boards say nothing about good play, so we also solved endgames exactly. Each random position was solved twice, once under each rule set, with no depth limit. The full study is in Is Quarto solved?. This is the headline:
Share of random endgames that are draws with perfect play
Squares also use up safe pieces faster. With 8 pieces placed, the player handing over a piece had no safe choice in 10.0% of random positions under classic rules and 27.5% with squares.
Classic Quarto from the empty board is a draw with perfect play (Luc Goossens, 1998). Our study doesn't re-solve the full game from the empty board with squares, so whether the square variant is also a draw with perfect play is not something these numbers answer.
In real games, squares decide most wins
We recorded which shapes completed the winning move in games of the Quads AI against itself, all played with squares:
| Winning move completed… | Grandmaster (2,891 wins) | Skilled (3,260 wins) |
|---|---|---|
| A 2×2 square (alone or with a line) | 54.2% | 54.4% |
| Only a 2×2 square | 42.6% | 42.0% |
| A row | 25.9% | 22.6% |
| A column | 30.8% | 35.9% |
| A diagonal | 3.0% | 2.0% |
Shares add up to more than 100% because 13.4% of Grandmaster's winning moves completed two kinds of shape at once. 43% of its wins came from a square that no line rule would have counted. In other words, under classic rules most of these games would have gone on.
Diagonals almost never decide these games. Both diagonals run through the centre, which the AI fills early, so they're usually dead long before the end.
Playing with squares
- Check 19 shapes, not 10. The most common mistake for players coming from classic Quarto is handing over a piece that completes a square nobody was watching.
- The centre is crowded. Each centre cell sits in 7 shapes. A careless piece there can open several threats at once, and so can a careful one, for you.
- Expect fewer draws. With squares, safe pieces run out sooner, so the middle game decides more. Count the safe pieces before every hand-off from about the sixth placement.
The strategy guide covers counting threats, and the Quads rules show all 19 shapes.
Frequently asked questions
What is the 2x2 square rule in Quarto?
A variant in which four pieces sharing at least one trait also win when they fill a 2x2 block of adjacent cells. It adds 9 winning shapes to the 10 rows, columns and diagonals.
How many winning shapes does Quarto have with squares?
19: 4 rows, 4 columns, 2 diagonals and 9 blocks of 2x2.
Does Quads use the square rule?
Yes. Quads always counts 2x2 squares as winning shapes, in every mode.
Does the square rule make Quarto harder?
It makes it sharper. Random boards almost always contain a winning shape with squares (all but 0.31%), solved endgames are draws far less often, and there are more shapes to check before every hand-off.