Cognivo

Learn Hex

Join your two edges. It can never end in a draw.

0 of 4 steps done

Step 1 of 4

Watch

Six neighbours

Every cell touches six others, so there are no corners to squeeze through.

The story

Invented Twice, Six Years Apart

Piet Hein invented Hex in 1942 and John Nash invented it again, independently, in 1948. Parker Brothers published it as Hex in 1952.

The whole story

Piet Hein introduced Hex at the Niels Bohr Institute in 1942 and published it in the Danish paper Politiken under the name Polygon. Six years later, and without knowing any of that, John Nash invented the same game again at Princeton. Students there called it Nash — or, less kindly, John, after the tiled bathroom floors they played it on. Parker Brothers sold it as Hex in 1952, which is the name that stuck. It is one of the few games with two credible independent inventors, and that is not a coincidence: the rules are close to the shortest possible description of a connection game, so anyone looking in that direction tends to find the same thing.

Why It Cannot End In A Draw

A filled Hex board always has exactly one winner. That is a theorem, not a rule, and it means blocking your opponent and connecting your own edges are the same move.

The whole story

Fill a Hex board completely, however you like, and exactly one player will have connected their edges. Not at least one — exactly one. It is impossible to fill the board and leave nobody connected, and it is impossible for both to connect at once. This is not a rule anyone imposed; it is a theorem, and it turns out to be equivalent to the Brouwer fixed-point theorem from topology. The practical consequence is what matters at the board: there is no such thing as a defensive draw. Blocking your opponent and connecting your own edges are the same act, because the only way to stop them is to build a chain of your own across their path.

The First Player Wins — And Nobody Knows How

The first player provably wins, by a strategy-stealing argument that gives no hint of the actual strategy. For an eleven by eleven board nobody knows what it is.

The whole story

There is a proof that the first player wins with perfect play, and it tells you nothing about how. It works by contradiction: suppose the second player had a winning strategy. The first player could simply make an arbitrary first move, then follow that strategy as though they were second. The extra stone can never hurt — stones are never captured and never in the way — so the first player would win too, and both players cannot have winning strategies. So the second player has none. That is a strategy-stealing argument, and it is entirely non-constructive. For boards of 11×11 the actual winning line is unknown. Which is why the swap rule exists.

The full notes

Everything above, and more, in words, to read or to listen to.

Rules3 sections

Join Your Two Edges

The board is a rhombus of hexagons. Two of its sides are red and two are blue. Players take turns placing one stone of their colour on any empty cell. Red wins by building an unbroken chain of red stones joining the two red sides; blue wins by joining the two blue sides. Stones are never moved, never captured, and never removed. Red plays first. That is the whole game.

  • Check which two sides are yours before your first move — it is the commonest beginner mistake.
  • A chain may wander anywhere; it does not have to be straight.

Six Neighbours, Not Eight

Each hexagon touches six others. That is what makes the game work, and it is worth a moment because a square grid would not. On a square board, two paths crossing diagonally can pass through each other — so both players could complete a connection, and a draw would be possible. Hexagons have no diagonals. Two chains either meet or they do not, and that is exactly why exactly one player always connects.

  • Two cells touching at a point only is not a connection — hexagons share edges.
  • The corner cells are the odd ones out: two of them touch only two others.

The Swap Rule

Because the first player wins with perfect play, Hex is played with a balancing rule. After red places one stone, blue may either answer normally or take that stone, becoming red and leaving red to move. This is sometimes called the pie rule, after the childhood method of one child cutting the cake and the other choosing a slice: it makes red's opening a genuine decision, because an opening too strong will simply be taken. Red's real task on move one is to find a stone that is not worth stealing. On Cognivo the swap is offered as a move on red's stone — click it.

  • Central openings are usually taken. Openings near the short corners usually are not.
  • With the swap on, opening strongly is a mistake, not an advantage.
Strategy4 sections

The Bridge

Two stones placed a short step apart, with two empty cells touching both of them, are already connected — even though they are not touching. If your opponent takes one of the two cells, you take the other. They cannot have both. This is called a bridge, and learning to see it is the single biggest jump in Hex strength. It lets you cross the board in half the stones, and it means a chain that looks full of gaps may be unbreakable. Most strong play is the construction of bridges rather than solid chains.

  • Count a bridge as connected. Filling it in yourself wastes a move.
  • Attacking a bridge is almost always pointless — answer on the other cell and nothing is gained.

Ladders, And Escaping Them

When you push toward your edge and your opponent blocks alongside, and you push again and they block again, the pair of you run a ladder across the board. Ladders are forced sequences, and the player running them gains nothing by continuing — the outcome was decided before the first step. What decides a ladder is whether there is a friendly stone waiting where it ends. A single stone placed far away, long before the ladder starts, can make the whole sequence win instead of lose. This is called a ladder escape, and it is why strong players place stones that appear to do nothing.

  • Work out where a ladder ends before you start one.
  • A stone doing nothing now may be the reason a ladder five moves later works.

The Short Diagonal

The centre is strong for the ordinary reason — it touches more of the board — but Hex has a second and more specific idea. The shortest line between your two edges runs along one diagonal of the rhombus, and it is the natural spine of your connection. Your opponent's spine is the other diagonal, and the two cross in the middle. That crossing is why the centre cell is the strongest opening, why it is always taken under the swap rule, and why the fight in a serious game happens in a band across the middle rather than near the edges.

  • Your short diagonal is your spine; your opponent's crosses it.
  • Edge play early is usually wasted — the game is decided in the middle band.

Blocking Is Building

New players tend to alternate between two modes: a few moves spent connecting, then a panicked move spent blocking. In Hex that distinction does not exist. The only way to stop your opponent crossing is to put a chain of your own across their path — and that chain is your connection. So the useful question is never "should I attack or defend". It is "which of my moves does most for my own chain while standing where theirs wants to go". If a move only blocks, and builds nothing, there is almost always a better one nearby.

  • A purely defensive move is a wasted move. Find the one that does both.
  • If you cannot see how a blocking stone helps you, look one cell over.