Short answer: under classic Quarto rules there are only two kinds of cell. The 4 corners and the 4 centre cells each sit on 3 winning lines, and a symmetry of the board turns any one of them into any other. The 8 edge cells sit on 2 lines. Add the 2×2 square rule and the centre splits off: each centre cell is in 7 winning shapes, against 4 for every other cell. We counted every shape and searched every symmetry of the board to check this.
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.
Lines through each cell
Classic Quarto wins on 10 lines: 4 rows, 4 columns and 2 diagonals. Every cell is in one row and one column. The diagonals pass through 8 cells, the 4 corners and the 4 centre cells, so those cells are in 3 lines and the 8 edge cells are in 2.
Quads, and the 2×2 variant that many Quarto editions include, also counts the 9 blocks of four adjacent cells. A corner is in 1 of those blocks, an edge cell in 2 and a centre cell in 4. That changes the picture:
| Cell type | Cells | Classic lines | 2×2 squares | Total shapes (with squares) |
|---|---|---|---|---|
| Corner (A1, D1, A4, D4) | 4 | 3 | 1 | 4 |
| Edge (B1, C1, A2, D2, A3, D3, B4, C4) | 8 | 2 | 2 | 4 |
| Centre (B2, C2, B3, C3) | 4 | 3 | 4 | 7 |
Classic rules: lines through each cell
With 2×2 squares: winning shapes through each cell
Why corners and centres are the same cell
Counting lines suggests corners and centre cells are alike under classic rules. Symmetry proves it. A symmetry of the board is any way to rearrange the 16 cells so that every winning line still lands on a winning line. Any position can be rearranged this way without changing anything about who wins.
We searched every possible rearrangement and found 32 symmetries of the classic board. Eight are the familiar rotations and reflections. The other 24 come from two less obvious moves, which can be combined with those:
- Swap the middle. Exchange rows 2 and 3 and, at the same time, columns B and C. Every row stays a row, every column a column, and both diagonals map onto themselves.
- Turn it inside out. Swap rows 1 and 2, and rows 3 and 4, and do the same with the columns. Corner A1 lands on centre B2, and the diagonals still map to diagonals.
The inside-out move is the one that matters. It sends every corner to a centre cell and back, so under classic rules there's no such thing as "taking the centre". A piece in the centre has exactly the same prospects as a piece in the corner.
Squares break this. Swapping the middle rows splits up 2×2 blocks, so once squares count, only the 8 rotations and reflections survive (our search found exactly 8). The corners, edges and centre become three different kinds of cell.
The first move has two real options
Put the symmetries together with the pieces and the opening shrinks to almost nothing:
- The first piece handed over doesn't matter. Relabel any trait, say swap "dark" and "light" on every piece, and every group that shared a trait still shares one. With those relabelings any piece can be turned into any other, so all 16 first hand-offs lead to equivalent games.
- The first placement has 2 real options under classic rules: a corner-or-centre cell, or an edge cell.
- With 2×2 squares it has 3: corner, edge or centre.
That's why opening theory in Quarto is thin, and why games are decided in the middle. With perfect play, classic Quarto is a draw (Luc Goossens, 1998, per Wikipedia), whichever of the two first placements you choose.
Where games are actually won
Geometry is one thing. To see where wins happen in practice, we recorded the cell of the winning placement in 4,000 games of the Quads Grandmaster AI playing itself, and 4,000 games at Skilled level. All games used 2×2 squares, and the first two turns were random so the games vary.
Grandmaster vs Grandmaster: where the winning piece was placed (2,891 won games)
| Cell type | Grandmaster wins | Skilled wins |
|---|---|---|
| Corners (4 cells) | 3.8% | 2.4% |
| Edges (8 cells) | 96.2% | 97.6% |
| Centre (4 cells) | 0.0% | 0.0% |
The centre took no winning pieces at all, even though it's in the most winning shapes. That isn't because the centre is weak. The AI fills the centre first, so those cells are occupied long before any line can be finished. Wins then land where the last gaps are: the edges, which took 96% of Grandmaster's wins.
One caution: these are games between copies of our AI, which has a strong habit of playing centrally. People open differently, so treat the heatmap as a picture of one style of play, not a law of the game.
What to do with this
- Classic rules: don't fight over the centre. It's equivalent to a corner. Think about which lines a placement opens or closes, not where on the board it is.
- With squares, a centre cell touches 7 shapes. A piece there can open, or kill, up to seven threats at once. Filling the centre early with pieces that share little with each other kills a lot of shapes at once and calms the game down.
- The last gaps decide the game. Late in the game, the empty edge cells are where your opponent's winning placement will come from. Check every shape through them before you hand a piece over. The strategy guide shows how.
Frequently asked questions
What is the best first move in Quarto?
Under classic rules there are only two genuinely different first placements, a corner-or-centre cell or an edge cell, and the first piece handed over makes no difference. Classic Quarto is a draw with perfect play whichever you choose.
Is the centre the strongest square in Quarto?
Not under classic rules: a symmetry of the board swaps every centre cell with a corner, so they are equivalent. With the 2x2 square rule, each centre cell is in 7 winning shapes against 4 for every other cell.
How many symmetries does the Quarto board have?
Thirty-two under classic rules. Only the 8 rotations and reflections remain when 2x2 squares also count.
How many winning lines go through each cell?
Corners and centre cells are on 3 lines (row, column, diagonal); edge cells are on 2. With 2x2 squares, corners and edges are in 4 shapes each and centre cells in 7.