Connect Four is Solved — How Victor Allis Cracked the Game

Connect Four is a “solved game” — mathematicians have completely figured out the outcome of perfect play. In 1988, Victor Allis proved that the first player can always win if they play optimally. Here’s what this means, how he did it, and why you should still enjoy playing.

What Does “Solved” Mean?

In game theory, a solved game is one where the outcome of perfect play has been calculated for every possible position. There are three levels of “solved”:

1. Ultra-weakly solved: We know who wins from the starting position with perfect play
2. Weakly solved: We know the outcome from the starting position AND have an algorithm to achieve it
3. Strongly solved: We know the optimal move from ANY possible position in the game

Connect Four is strongly solved. For every one of the approximately 4,531,985,219,092 possible positions, mathematicians have computed:

• Whether the current player can force a win
• The exact move the current player should make
• How many moves until the game ends with optimal play

How Victor Allis Solved Connect Four

In 1988, Dutch mathematician Victor Allis solved Connect Four as part of his research at Vrije Universiteit Amsterdam. His approach combined several techniques:

Knowledge-Based Approach

Rather than brute-force checking every position (computationally expensive), Allis developed strategic rules that capture expert human knowledge. These rules identify patterns like:

• Threats (three in a row with an open fourth slot)
• Double threats (forcing moves)
• Zugzwang positions (any move loses)
• Column control advantages

This knowledge-based approach dramatically reduced the search space by pruning branches where the outcome was already determined.

Exhaustive Database

For positions where strategic rules couldn’t determine the outcome, Allis built a retrograde analysis database — solving backwards from won positions to determine which earlier positions lead to wins.

The final database required approximately 300 KB of storage (tiny by modern standards) and could evaluate any position in milliseconds.

Key Findings

Allis’s proof established several facts:

1. First player wins if they start in the center column and play perfectly afterward
2. Draw is possible if the first player opens elsewhere or if the second player defends perfectly against non-optimal first moves
3. Center column is the ONLY winning first move — all other opening moves can be held to a draw by perfect defense

Independent Verification

Shortly after Allis’s proof, James D. Allen independently solved Connect Four using pure brute-force computation. His method stored evaluations for all positions directly, requiring more computing power but arriving at the same conclusions. This independent verification strengthened confidence in the solution.

The Mathematics Behind the Solution

Game State Space

The standard 7×6 Connect Four board has:

The game tree (all possible sequences of moves) is vastly larger — approximately 10^21 (1 sextillion) unique games can be played.

Position Evaluation

Each position in Connect Four falls into one of three categories:

1. Winning position: The current player can force a win
2. Losing position: The opponent can force a win regardless of play
3. Drawing position: Perfect play by both sides leads to a draw

Allis computed not just win/lose/draw but also the distance to forced outcome — how many moves until the game ends with optimal play from each side.

Position Types by First Move

Here’s what perfect play produces based on the first player’s opening move:

Only the center column opening guarantees a win for the first player. This is why control of the center is the most important strategic principle in Connect Four.

Why Perfect Play is Impossible for Humans

If Connect Four is solved, why do humans still lose and win against each other? Because perfect play requires:

1. Impossible Memorization

The complete solution database contains optimal moves for ~4.5 trillion positions. Even if you memorized one position per second, it would take over 140,000 years to learn them all. No human can memorize even a tiny fraction of this.

2. Deep Calculation

Without memorization, you’d need to calculate 15-20+ moves ahead in many positions — something grandmaster chess players struggle to do in a simpler game tree. The branching factor in Connect Four (7 possible moves per turn) compounds to millions of positions quickly.

3. Pattern Recognition Limits

Experienced players recognize key patterns (threats, traps, control positions) but miss subtle interactions. A computer sees everything; humans see heuristics.

Human vs. Computer Performance

This is why Connect Four remains fun — the solution exists in theory, but in practice, every game is unpredictable.

Lessons from the Solution

The solved nature of Connect Four teaches us important strategic principles:

1. The Center Column is Everything

The mathematical proof confirms what good players intuit: controlling the center column provides maximum winning potential because it connects to more possible four-in-a-rows than any other column.

2. First Player Advantage is Real but Fragile

Yes, the first player can always win — but only with perfect play starting from the center column. Any mistake converts the game to equal or worse. The advantage exists but exploiting it requires superhuman precision.

3. Most Games are “Solved” at Birth

In retrospect, most Connect Four games are theoretically decided in the first few moves. If both players opened with perfect moves, we could predict the outcome from move 4 or 5. Of course, neither player is perfect, so the actual outcome depends on who makes fewer (or less critical) mistakes.

4. Odd/Even Strategy Matters

One finding from Allis’s analysis: Whoever controls the odd-numbered rows (1, 3, 5) often controls the game. This is because the first player gets the odd moves (1st, 3rd, 5th…), and certain winning threats require landing on specific rows. Expert players track parity carefully.

Other Solved Games

Connect Four joins a family of games where perfect play has been computed:

Connect Four sits in a sweet spot — complex enough to be interesting, simple enough to be fully computed.

Can Computers Beat You at Connect Four?

Yes, easily. Modern Connect Four solvers play perfectly because they have access to the complete solution database. When you play against a “Perfect” difficulty AI, you are literally playing against the solved game.

However, most online implementations offer adjustable difficulty:

• Easy: AI makes intentional mistakes
• Medium: AI plays well but not optimally
• Hard: AI plays near-perfectly
• Perfect: AI plays the mathematically optimal move every time

Against perfect play, the only ways not to lose are:

1. Play first and execute perfect strategy (draw or win)
2. Force a draw from a drawing position
3. Get lucky that the AI isn’t actually perfect

Most humans lose to perfect play, but that’s fine — playing against humans is where the fun happens.

Playing Connect Four Knowing It’s Solved

Does knowing the solution ruin the game? Not really. Here’s why:

The Joy is in Human Competition

When you play another person, you’re competing against their imperfect memory and calculation — just like they’re competing against yours. The outcome depends on relative skill, not absolute perfection.

Strategic Depth Remains

Understanding general principles (center control, threat creation, avoiding traps) is more important than memorizing positions. Great players win not by knowing solutions but by applying principles better than opponents.

The Solved Status Validates Strategy

Knowing that center control and first-player priority are mathematically verified makes learning strategy more satisfying. You’re not guessing at best practices — you’re applying proven principles.

Conclusion

Connect Four was mathematically solved by Victor Allis in 1988, proving that perfect play from the first move leads to a win for the first player (via the center column) or a draw (from any other opening). This makes Connect Four one of the most complex games ever fully computed.

Yet the game remains endlessly playable. No human can memorize 4.5 trillion positions or calculate 20 moves deep on every turn. The solution exists in theory while uncertainty thrives in practice. Every game you play is a contest of skill, intuition, and luck — just like before we knew the perfect answer.

So go ahead — play Connect Four. Start in the center column. Apply the principles you’ve learned. And remember: your opponent doesn’t know the solution either.