Connect Four Endgame Strategy - Counting Squares and Winning the Zugzwang Battle

Most Connect Four articles stop at the middlegame. Build a fork, block a fork, control the center.

Then the board fills up, both players have blocked everything, and something strange happens. Neither side has a move they want to make. Both sides have to move anyway. One of them runs out of safe squares, drops a disc where they never wanted to, and loses a game they thought was drawn.

That is the endgame. It is not about finding a clever move. It is about counting.

You Cannot Pass

Chess players call it zugzwang: the compulsion to move. You are fine as long as nobody makes you play, and the moment you have to play, your position collapses.

Chess at least gives you dozens of quiet moves to shuffle around with. Connect Four gives you seven columns, and every single one fills up. There is no pass, no null move, no way to hand the turn back. Every turn you must put a disc somewhere, and every disc you put down opens the square above it.

That last part is the whole game. In Connect Four you never just occupy a square. You also hand your opponent the square on top of it.

Once the board is mostly full, that becomes the only thing that matters.

Every Empty Square Has an Owner

Late in a game, stop looking at discs and start looking at holes.

Take each empty square still on the board and ask three questions:

1. If you filled it, would you have four in a row?
2. If your opponent filled it, would they have four in a row?
3. If neither, it is a dead square.

That gives you a map. Winning squares for you, winning squares for them, and dead squares that belong to nobody.

Dead squares are the currency of the endgame. They are the moves you can make without giving anything away. Every dead square you can still play into is one more turn you survive. When you run out of them, you are forced into a live column, and the map tells you exactly what happens next.

Most players never build this map. They keep hunting for an attacking idea in a position that has been decided for ten moves. The stronger habit is boring and mechanical: count the holes, label them, count again.

The Column Is the Unit of Counting

Here is the rule that makes the whole thing tractable.

If both players do nothing but fill a single column until it is full, that column absorbs exactly as many moves as it had empty squares.

So:

• A column with an even number of empty squares gives the turn back to whoever entered it first. Parity is preserved.
• A column with an odd number of empty squares hands the turn to the other player. Parity flips.

That is your lever. Every open column is either a parity-preserving column or a parity-flipping column, and the player who understands which is which controls who gets forced into the poisoned column at the end.

There is an even more direct version of this for a single threat. The board has 42 squares. Red plays moves 1, 3, 5 and so on, Yellow plays the even ones, so on a board that fills completely the 42nd and final disc is always Yellow's.

Now pick any threat square. Count the empty squares directly underneath it in its own column, add one for the threat square itself, and add the number of discs already on the board. If that total is odd, the square lands on Red's turn. If it is even, it lands on Yellow's. Then adjust for every disc that gets played in another column before you get there.

That is the arithmetic behind the odd/even threat rule. Odd rows tend to favor Red and even rows tend to favor Yellow because of where the count usually lands. But the rows are a shortcut. The count is the truth, and in a real endgame with three columns still open you should do the count.

A Position Where Whoever Moves Loses

Enough theory. Here is a position with eight empty squares and only two open columns, verified against the perfect solver.

Map the holes first. Rows are numbered from the bottom, so row 3 is the next square to land in each open column.

Column 1, bottom to top:

• Row 3: dead
• Row 4: Red wins (it completes the row-4 line across columns 2, 3 and 4)
• Row 5: dead
• Row 6: Yellow wins (it completes the row-6 line across columns 2, 3 and 4)

Column 7, bottom to top:

• Row 3: dead
• Row 4: Red wins (it completes the diagonal from column 4 row 1 up to column 7 row 4)
• Row 5: dead
• Row 6: dead

Now look at what that means.

Yellow can never move here. Yellow's only options are row 3 of column 1 or row 3 of column 7, and both of them put Red's winning square directly in reach. Yellow drops, Red drops on top, game over in two plies.

Red cannot move here either. Red plays row 3 of column 7, Yellow is forced to take row 4 to stop the diagonal, and now rows 5 and 6 of column 7 are dead squares that both players burn through. Column 7 empties out. Red has nothing left but column 1, plays row 3, Yellow blocks row 4, Red plays row 5, and Yellow drops the 42nd disc on row 6 to win on the top row.

Both sides are lost. The only thing that decides the game is whose turn it is.

Notice why Red cannot escape by burning column 7 first. Column 7 has four empty squares, an even number, so filling it returns the turn to Red. Red enters it, and Red comes out of it. The spare moves are real but they are useless, because there is an even number of them.

Flip One Disc, Flip the Result

Take the same position and give Red one extra disc at the bottom of column 7.

Same threats. Same winning squares. Same dead squares. The only change is one disc, and now the perfect solver scores it as a win for Yellow.

Yellow takes row 4 of column 7 immediately, killing Red's diagonal, and from there Red is the one who runs out. Red is forced into column 1, Yellow blocks row 4, Red has to play row 5, and Yellow takes row 6 on the top row again.

One disc. Total reversal. That is how thin endgame Connect Four is, and it is why counting beats intuition once the board is three quarters full.

How to Count in a Real Game

You will not have a solver at the board. You will have about thirty seconds. Here is the practical sequence.

Find the live squares. Scan every open column for a square that completes four for either player. Usually there are only two or three on the whole board. Mark them.

Work out who owns each one. Count the empties below it, plus one, plus the discs on the board. Odd means Red, even means Yellow. This is fast once you have done it a few times.

Count your dead squares. Add up every square you can play into without opening a live square for your opponent. That number is how many turns you can survive.

Count theirs. Same thing from their side. Whoever has more spare moves wins the waiting game.

Check the parity of each open column. Even columns keep the turn, odd columns flip it. If you are one move short, look for an odd column that fixes it.

Then decide the order. You almost always have a choice about which spare column to burn first. That choice is the endgame. Burn the odd column when you need to flip the turn, burn the even one when the count is already in your favor.

Four Endgame Mistakes That Lose Drawn Games

Playing directly under your own winning square. This is the most common blunder in the whole game. You have a winning square on row 4. You play row 3 yourself. Your opponent immediately takes row 4 and your threat is gone. If a square wins for you, somebody else has to be the one who opens it.

Burning your spare moves early. Those quiet dead squares are not filler. They are the only reason you will still be alive in six moves. Playing them out of boredom in the middlegame is handing the opponent the endgame before it starts.

Assuming a blocked threat is a dead threat. In the position above, Red's row-4 square in column 1 never disappears. It sits there forcing Yellow to spend a move blocking it at exactly the wrong moment. A threat your opponent has to answer is still doing work even if it never scores.

Counting rows instead of counting squares. "Odd rows are mine" is a fine heuristic in the middlegame. In an endgame with three open columns of different heights, it will get you killed. Count the actual squares.

Where This Fits

The endgame is the last of three layers, and they stack.

The opening decides which threats are even available. Column 4 is the only first move that wins with perfect play, and that is a solved fact, not an opinion.

The middlegame decides where the threats land. That is double threat territory: build two live lines, force your opponent to keep answering, and put your threats on squares whose parity favors you.

The endgame decides who reaches those squares. That is counting, spare moves, and zugzwang. If you want the compressed definition, the zugzwang glossary entry and the tempo entry cover the vocabulary.

For the second player specifically, this is the layer that turns a losing game into a draw. Most of the P2 draw strategy is really an endgame counting plan disguised as defense: survive the middlegame, keep one counter-threat alive, and make sure the last forced move belongs to Red.

Practice

Endgame counting is the most learnable skill in Connect Four. It is arithmetic, not talent.

Start with puzzles. The harder ones are endgame positions where the winning move is a quiet square in an unrelated column, and finding it is exactly the counting drill described above.

Then play the engine and refuse to resign a locked position. Play it to the last disc. The engine will punish every parity error you make, and it will do it the same way every time, which is how you learn the pattern.

After that, review your games. Look specifically at the last ten moves of any game you drew or narrowly lost. Count the spare moves at move 30 and see who was actually ahead. Most players discover their losses were settled long before they noticed.

No pass. No filler. Every disc you drop opens something for somebody.

Count the holes, and the endgame stops being luck.