Hex board game

The rules of Hex, and the swap rule this table leaves out

Each player owns two opposite edges of the grid, which is a rhombus of hexagons, eleven on a side. Jade takes top and bottom, and places the first stone. Ivory takes left and right. On your turn you put one stone of your colour in any empty hexagon, and every empty hexagon is legal. Stones never move, never capture and never come off. You win the moment an unbroken chain of your own stones touches both of your edges. That is the whole of the rules, and the only other thing you can do at this table is resign.

That puts it on the abstract strategy board game shelf, among the games with no dice, no cards, no hidden information and nothing to memorise. It is a strategy game in the plainest sense: one hundred and twenty-one hexagons, one stone a turn, and nothing on the board ever changes hands.

One rule you will meet everywhere else is not on this table. Whoever opens at Hex has an advantage, and the swap rule, also called the pie rule, is the usual answer to it: the second player looks at the opening move and may choose to take it as their own, which pushes the first player to open with something they would not mind facing.

This table does not implement it. Both sides place in strict alternation here, from the first stone to the last, with no swap, and the file that runs it has exactly two actions in it: place a stone, or resign. Most of what you will read about Hex elsewhere assumes the swap rule is in force, so the version here is not the version those pages describe. Play two games instead and let the other person open the second one. That costs nothing and settles the same complaint.

Which Hex this is

Type the name on its own and the internet hands you hexadecimal, colour codes and a data company. There is also a digital trading card game called Hex, its cards long switched off, and a video game called The Hex. Search for a hex board game cafe and you land somewhere else again, at a games cafe trading under the same short word. None of them is the board set out above.

This page is the hex board game, published in 1942. It belongs to the connection games, the family that also holds Y and Havannah: you win by joining two edges, not by taking anything. If you arrived from a search that returned one of the others, the rules above are the ones that apply here, and nothing written about the card game or the video game touches them.

Two mathematicians, six years apart

Piet Hein invented it in 1942 at the Niels Bohr Institute in Copenhagen, and the Danish newspaper Politiken printed it on 26 December of that year under the original name Polygon. He later called it Con-tac-tix. In 1948 or 1949 John Nash arrived at the same game at Princeton, where his colleagues called it Nash, or sometimes John, because it could be played on the hexagonal tiles of a bathroom floor. Parker Brothers published it as Hex in 1952, on the eleven by eleven board that is still standard, and sold a Con-tac-tix edition in 1968. Hex is the name that stuck.

The two inventions are usually reported as independent, and that is how Nash described his. The point is not settled. There were Danish players at Princeton in the 1940s, which leaves a route by which the game could have reached him. Hein’s date is attested in print. Nash’s independence is his own account, and a reasonable doubt sits beside it. Eighty years on, that is as close to the truth as the record gets. Between them they left the game four names in ten years: Polygon, Con-tac-tix, Nash and Hex.

The board game that cannot be drawn

Fill the board completely, in any order, by any two people playing as badly as they like, and exactly one player will have joined their edges. Not sometimes. Always, and only one of them. There is no drawn position in Hex, no stalemate and no repetition.

Set that beside the games it is usually compared with. Chess can be drawn, and at the highest level it is drawn often. Go can be tied when the komi is a whole number. Hex can do neither, at any board size.

It changes what a bad position feels like. In chess you can dig in and play for the half point, and in Hex there is no half point to play for, so a losing game has to be turned round or lost outright. Losing here is blunt for that reason, which is the honest thing to tell somebody sitting down to their first one. The compensation is that nothing is dead until the connection is finished, because your opponent still has to finish it, and every cell they have not filled yet is a cell you can still reach first.

Mathematicians took the fact seriously enough to work out what it rests on. In 1979 David Gale published a proof that the determinacy of Hex is equivalent to the two dimensional Brouwer fixed point theorem, so the reason your last empty hexagon has to decide something is the reason a stirred cup of coffee always has one point on its surface that ends where it started.

The first player wins, and nobody knows how

Here is the part the ranking pages state and then walk away from. The player who moves first has a winning strategy. The original argument is Nash’s, from around 1949, and it is called strategy stealing: suppose the second player had a winning strategy, then the first player could play anywhere at all, adopt that strategy as their own and be one spare stone ahead the whole way. In Hex an extra stone of your own colour is never a liability, so the assumption collapses and the first move has to be the winning one.

Now the part that usually goes missing. Every known proof of this is non-constructive. It proves that a winning strategy exists without producing a single move of it. The theorem is a promise with no instructions inside.

Machine searches have closed the gap only at the edges. Jakub Pawlewicz and Ryan Hayward solved every opening on the nine by nine board in 2013, along with the most central opening on ten by ten, and smaller boards had fallen before that. Stefan Reisch had already shown in 1981 that deciding Hex positions in general is PSPACE-complete, after Shimon Even and Robert Tarjan proved the same for generalised Hex on arbitrary graphs in 1976. The eleven by eleven board this table sets out has not been solved. So the honest statement is this: whoever opens on our board is theoretically already winning, and no person and no machine can tell them how.

Which leaves a fair objection from anybody meeting the game tonight. If the result is settled before the first stone goes down, why sit at the board at all? Because nobody can play the settled game. Across 121 cells neither the strongest player alive nor the strongest program holds that strategy, so what decides a real game is which of you sees the next cut first.

How to win at Hex

Because there are no draws, attack and defence are one move. Any stone that blocks your opponent’s path is also a stone on the way to your own edges, and any stone that connects you narrows the ground they have left. A game of Hex rarely feels like two people taking turns at separate projects, because both sides are building on the same hexagons.

The first real idea beyond that is the bridge. Two of your stones set diagonally apart, with two empty hexagons between them, are already connected even though nothing joins them yet: if your opponent takes one of the two gaps, you take the other. Play it once and it sticks. Say you are joining top to bottom. Put a stone in the middle of the board, then put the next one two rows on and one column back. Exactly two empty hexagons touch both stones. Your opponent can have either of them and never both, so the pair is already joined and you have crossed two rows for the price of one stone. Stringing bridges is how a strong player crosses the board in half the stones a beginner needs.

The computer here has never been told what a bridge is. It scores every empty cell by how many stones each side would still need to join their edges, then takes the one that shortens its own road and lengthens yours, after first checking whether anything wins outright or stops you winning outright. It is a shortest path player rather than a deep searcher, it leans very slightly towards the middle of the grid and towards its own stones when nothing else separates two moves, and it will punish a loose chain long before you notice you left one.

Tactics past the bridge

The word the game itself uses is safely connected. A group of stones is safely connected when no reply your opponent can make will separate it, and the bridge is the smallest pattern that qualifies: two stones of one colour and a pair of open spaces, either of which does the job on its own. Playing Hex well is building small safely connected patterns and then joining those patterns into a path, rather than laying one stone against the next.

That is the correction most beginners need. The instinct is contact, putting your stone against your opponent’s, and contact is usually a wasted move here. Connection beats contact. A chain with gaps you can always fill travels twice as far as a solid wall, and a wall costs you the space you needed for the rest of the crossing.

One opening difference matters on this table. Where the swap rule is in force, the first player has to open somewhere modest, because anything too strong is taken off them. With no swap here that pressure is gone, and the middle of the board is there for whoever opens. The centre of an eleven by eleven board is the sixth hexagon of the sixth row, and it is the opening the swap rule was invented to discourage. On the ten by ten board the only opening anybody has solved is the most central one, and it is a first player win.

Board sizes, and why this one is eleven

Eleven by eleven is the standard, and it is the size Parker Brothers put in the box in 1952. It is not the only one played. Thirteen by thirteen and nineteen by nineteen are both popular, and the 19x19 comes straight from Go, whose board that is. A Beautiful Mind reports that Nash argued for fourteen by fourteen as the best size. Nestorgames publishes the game today in eleven, fourteen and nineteen.

Size changes two things and no others. A bigger game board is more space, a longer game and more room for a plan to come apart in the middle of it. Eleven by eleven is 121 hexagons, thirteen by thirteen is 169, and nineteen by nineteen is 361, which is roughly three times the ground to cross and three times the stones to cross it with. A smaller one is closer to being solved: every opening on nine by nine has been worked out, and the seven by seven has an explicit winning strategy written down. Nothing about the rules changes at any size, and the first player still theoretically wins at all of them.

There is also nothing to buy afterwards. Hex has no expansion, no second edition with better components, no cards to add. The game board is the whole game. Related games exist, and they are separate games rather than add-ons: Rex, which is Hex played to lose, along with Y, Havannah, Projex and Dark Hex, where you cannot see your opponent’s stones.

A machine made of light bulbs

Around 1950 Claude Shannon and E. F. Moore built a machine that played Hex. It was an analog device: a resistance network, with resistors standing in for the connections between cells and light bulbs for the cells themselves. Current found the easiest way across the board, and the machine played where the current most wanted to go. A game about joining two sides turns out to be a game about electricity looking for a path.

Programs took a long time to get good at it. Monte Carlo tree search took over from about 2006, and on 30 October 2019 a program called Mootwo, built on Polygames with AlphaZero style self learning and fully convolutional networks, beat the highest ranked human player on LittleGolem. The Computer Olympiad has run Hex for years, with the pie rule in force.

None of that produced the thing the theorem promises. Machines have been beating the strongest people alive at this game since 2019, and the winning strategy the strategy stealing argument guarantees on an eleven by eleven board is still nobody’s to write down.

Making a hex board yourself

Hex asks for almost no equipment: a hex grid and two ways to mark it. Making your own board is a sheet of hex paper and two pens. Draw a rhombus eleven hexagons on a side, then mark the top and bottom edges in one colour and the left and right edges in the other, so each side can see which two it is joining. Drawn at half an inch to a cell, an eleven by eleven board is around eight inches across, so the whole thing fits on one sheet. People also print it, 3D print the tiles, and scratch it on whatever is on the table, which is why the game keeps turning up handmade. In 1974 the 3M paper games series sold it as a fifty sheet pad of ruled Hex grids.

Go players tend to recognise it on sight, because the pieces behave the way Go stones do: placed once, then never moved again. A Go board is bigger and its stones can be captured, but the hand is the same hand. The kinship is close enough that Hex turns up on Go forums more often than on board game shelves.

Playing Hex online, with someone who is not in the room

One click here sets a board out and returns a link. Give it to whoever should hold the other two edges, and they are sitting opposite you the moment they follow it, with no account to make and nothing to install. If the two of you are already in the same room, a single screen turned between turns does the same work. If there is nobody to give it to, the computer takes the other colour.

Hex is public domain, which is why it is here at all. These boards are being rebuilt one at a time out of games nobody owns, so that people who are a long way from each other have something to sit down to. Hex is the modern classic at that end of the shelf, a strategy game whose rules fit in a sentence and whose play does not.

You do not need any of the rest of this first. Two edges, one stone a turn, no draws. Come and take a seat with us and work the rest out on the board, which is how Hein handed it to a newspaper in the first place.