The Quarto endgame is a race to run your opponent out of safe pieces. A piece is safe to hand over if your opponent can't win with it anywhere on the board. In 4,000 games of our strongest computer player against itself, the giver still had 7.3 safe pieces out of 12 after four placements, but only 4.2 of 8 after eight and 1.7 of 6 after ten. By then a third of positions had no safe piece at all. Win the endgame by making sure that happens to your opponent, not to you.
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.
How fast safe pieces run out
We played 4,000 games of the Quads Grandmaster AI against itself, and 4,000 at Skilled level, all with the 2×2 square rule. Before every hand-off we counted how many of the remaining pieces were safe.
Average number of safe pieces at each hand-off
Grandmaster games: positions where every piece loses
| Pieces left | Positions | Average safe | Share safe | No safe piece |
|---|---|---|---|---|
| 12 | 4,000 | 7.28 | 61% | 0.0% |
| 11 | 4,000 | 4.12 | 37% | 0.0% |
| 10 | 4,000 | 3.75 | 38% | 1.9% |
| 9 | 3,922 | 3.30 | 37% | 5.1% |
| 8 | 3,722 | 4.22 | 53% | 5.9% |
| 7 | 3,501 | 2.22 | 32% | 6.5% |
| 6 | 3,275 | 1.70 | 28% | 33.7% |
| 5 | 2,170 | 1.76 | 35% | 7.2% |
| 4 | 2,013 | 1.46 | 37% | 25.2% |
| 3 | 1,506 | 1.51 | 50% | 4.8% |
| 2 | 1,434 | 1.15 | 57% | 15.6% |
The drop is steepest early: from 12 safe pieces at the fourth hand-off to about 7 at the fifth and about 4 at the sixth. Strong players don't lose there, but that's where the endgame is set up. With six pieces left, a third of positions were already lost for the player handing over.
These are games between copies of our AI, not perfect play. For exact results on random positions under both classic and square rules, see Is Quarto solved?.
How to count safe pieces
Counting safe pieces sounds slow, but it reduces to one question per open line. Every trait splits the 16 pieces in half, so each open line rules out a half of the set:
- Find every shape with three pieces and one gap. Rows, columns, diagonals, and 2×2 squares if you play with them.
- For each, name what the three share. If they share nothing, the line is dead: forget it. If they share "short", any short piece wins there.
- A safe piece must break every one of those traits. Short threat? It must be tall. Light threat? It must be dark.
- Look for those pieces in the pool. If none is left, you're in trouble, and it's better to find that out one turn early.
The numbers behind step 2: of the 560 possible groups of three pieces, 208 share no trait (dead), 256 share one trait (5 of the other 13 pieces finish them) and 96 share two (9 pieces finish them). A line of three sharing two traits is twice as dangerous as one sharing one.
A worked example
This position comes from one of our Grandmaster games. Nine pieces are on the board and you must hand over one of these seven:
- Row 1 (A1, B1, D1) is all short, so any short piece wins at C1.
- Column B (B1, B2, B3) is all light, so any light piece wins at B4.
- The top-left square (A1, B1, B2) is short and light, so it adds nothing new: those pieces already lose.
A safe piece has to be tall and dark. Of the seven, only one is: the tall dark square solid piece. Every other piece loses on the spot. That's the endgame in miniature: two threats on different traits cut the pool to a quarter, and here the quarter holds exactly one piece. Hand it over, and your opponent now has to find a placement that doesn't leave you with an easy win.
Turning a shortage into a win
Counting keeps you alive. To win, turn it around: place your piece so that, after you hand over a safe piece, every placement your opponent can make leaves them nothing safe to give back. That's the squeeze, and our strategy guide walks through one move by move. Our Quarto puzzles include four squeeze positions to practise on, each checked by computer.
Endgame habits
- Count from the sixth placement. That's where safe pieces drop from most of the pool to a handful.
- Pick your hand-off before you place. A placement that looks harmless can make every remaining piece unsafe. Decide what you'll give, then check the placement keeps it safe.
- Kill lines on purpose. Placing a piece that shares nothing with a line's other pieces kills that line for good, and fewer live lines means more safe pieces for you.
Frequently asked questions
What is a safe piece in Quarto?
A piece your opponent cannot win with anywhere on the current board. Handing over an unsafe piece loses on the spot if they see the win.
How do you win the endgame in Quarto?
By leaving your opponent with no safe piece to hand back. Place your piece and choose your hand-off so that, wherever they put it, every remaining piece wins for you.
When do safe pieces run out in Quarto?
In our strong computer games, the giver had about 4 safe pieces out of 8 after eight placements, and in 34% of positions with six pieces left there was none.
How many pieces can finish a line of three?
Five of the remaining 13 if the three pieces share one trait, nine if they share two, and none if they share nothing.