Quarto in numbers: 16 pieces, 4 traits, a 4×4 board and 10 winning lines (19 if 2×2 squares count). Of the 1,820 ways to choose four pieces, 536 (29.45%) share a trait and would win if lined up. A board filled completely at random contains no winning line only 1.98% of the time. With perfect play the game is a draw (Luc Goossens, 1998). Every figure below that isn't cited was computed by us, and the method is at the bottom.
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 pieces
- 16 pieces. Four traits with two versions each (tall/short, dark/light, round/square, solid/hollow) give 24 = 16 combinations, and the set has exactly one piece for each.
- 8 pieces carry each trait value. There are 8 tall and 8 short, 8 dark and 8 light, and so on for all eight values. See the full chart of the pieces.
- 120 pairs of pieces. Of those pairs, 32 share three traits, 48 share two, 32 share one and just 8 share nothing. Those 8 pairs are each piece and its exact opposite.
- Every piece has exactly one opposite. Put a piece and its opposite in the same line and no one can ever win that line.
Lines and winning groups
- 1,820 ways to pick four pieces. That's 16 choose 4.
- 536 of them share at least one trait (29.45%). Line up four pieces picked at random and they form a "Quarto" a little under 3 times in 10.
- 512 share exactly one trait, 24 share exactly two, and none share three. Four different pieces can't agree on three traits, because only two pieces exist for any three fixed values.
- Each trait value makes 70 winning groups. Choose any 4 of the 8 dark pieces: 8 choose 4 = 70. Eight values times 70 is 560, minus the 24 groups counted twice, which gives the 536.
- 37.1% of three-piece lines are already dead. Of the 560 possible groups of three, 208 share no trait and can never be completed.
- A live line of three has either 5 or 9 winning pieces. If the three share one trait, exactly 5 of the other 13 pieces finish it. If they share two traits, 9 do. Across all groups of three the average is 3.83.
How many of the 13 other pieces complete a line of three?
That 5-or-9 split is the most useful number in the game. When three pieces in a line share one trait, more than a third of the 13 remaining pieces (5 of 13) are dangerous to hand over. When they share two, it's 9 of 13.
The board
- 16 cells, 10 winning lines: 4 rows, 4 columns and 2 diagonals.
- 19 winning shapes with the 2×2 square rule. The variant adds the 9 blocks of four adjacent cells. Quads always plays this way. The square variant guide explains what it changes.
- Every cell is in 2 or 3 lines. The 4 corners and 4 centre cells sit on a diagonal, so they're in 3 lines. The 8 edge cells are in 2.
- With squares, the 4 centre cells are in 7 shapes. Each corner and edge cell is in 4.
- The classic board has 32 symmetries. These are the ways to rearrange the 16 cells so every line stays a line. Rotations and mirror images give 8. The rest come from less obvious swaps, such as exchanging the two middle rows and the two middle columns at the same time.
- With squares, only 8 symmetries survive. They are the rotations and reflections of the square. The swaps that keep lines intact break up the 2×2 blocks.
Winning shapes through each cell (rows, columns, diagonals and 2×2 squares)
Random boards and random play
What happens if nobody thinks at all? We filled 1,000,000 boards with the 16 pieces in a random order.
- 1.98% of random full boards contain no winning line (95% interval 1.95–2.01%). A draw takes effort: chance alone almost never produces one.
- With 2×2 squares, only 0.31% do. That's about 1 board in 323.
- A random full board holds 2.9 winning lines on average, and 5.6 winning shapes once squares count.
- Random play first wins on placement 11.6 on average under classic rules, and 10.5 with squares. The earliest possible win is on the 4th placement.
When does random play first complete a winning shape?
Games between computer players
We also had the five computer opponents in Quads play each other: 10,000 games in total, 400 for every pairing and seat, all with 2×2 squares. The full results are in our AI level test.
- Grandmaster scored 81.8% against the other four levels, counting a draw as half a point. Rookie, which plays almost at random, scored 1.2%.
- Good games last longer. When Grandmaster played itself, games averaged 12.7 placements and 27.7% were draws. Rookie against Rookie averaged 9.11 placements.
- 54% of Grandmaster's wins against itself completed a 2×2 square. Under Quads rules the squares aren't a side rule. They decide most games.
Perfect play
- Classic Quarto is a draw with perfect play. Luc Goossens solved it by computer in 1998, according to Wikipedia's Quarto article, which also reports that the result was corroborated again in 2026.
- 20,922,789,888,000 ways to fill the board. That's 16! (16 factorial). Each of the 16 pieces can go in any of the 16 cells, with no repeats.
- Fewer than 437,763,136,697,395,060,000,000,000 possible games. That's (16!)2: one factor for the order the pieces are handed over, one for where they go. It's a loose upper bound, since most games end long before the board is full.
Want to see how the squares change perfect play? Our solver study, Is Quarto solved?, has the endgame numbers.
How we computed these
- Counts of pieces, groups, cells and symmetries are exact. We enumerated every case: all 1,820 groups of four, all 560 groups of three, all 120 pairs. The symmetries came from an exhaustive search over rearrangements of the 16 cells.
- Random boards are 1,000,000 uniformly random orderings of the 16 pieces into the 16 cells, from a fixed seed. Intervals are 95% Wilson intervals.
- Computer games use the unmodified Quads AI. The first two turns of each game are random, so the deterministic levels don't repeat the same game. These numbers describe our AI, not perfect play.
- Everything runs from one script with a fixed seed (20261028), so it reproduces exactly. Last run October 28, 2026.
Frequently asked questions
How many winning combinations are there in Quarto?
Of the 1,820 ways to choose four of the 16 pieces, 536 share at least one trait. On the board there are 10 winning lines, or 19 winning shapes if 2x2 squares count.
What are the odds that four random Quarto pieces share a trait?
29.45%: 536 of the 1,820 possible groups of four.
Can a Quarto board fill up with no winner?
Yes, but rarely by chance. Only 1.98% of boards filled at random have no winning line under classic rules, and 0.31% when 2x2 squares count. With perfect play, though, classic Quarto is a draw.
How many symmetries does a Quarto board have?
Under classic rules, 32 rearrangements of the cells keep every row, column and diagonal a winning line. When 2x2 squares also count, only the 8 rotations and reflections remain.