Bored with board games? If so,you should give the game ofHex a try. It’s just as addictive asthe best computer games, and it

gives your brain a far more stimulatingworkout. Cameron Browne, a computerprogrammer in Brisbane, Australia, hasrecently published Hex Strategy: Making theRight Connections (A. K. Peters, 2000), thefirst book to take a comprehensive look atHex and how to win it. The book is a mustfor every recreational mathematician.

Hex is a two-person game, played on aboard made from hexagonal cells arrangedin the shape of a rhombus [see illustra-tion on opposite page]. The standardboard is 11 by 11. Each player“owns” two opposite edges ofthe board, and the four cor-ner cells are joint property.Each player also has a stockof counters; in the illus-trations here, one playeruses red counters andthe other uses blue. Therules are astonishinglysimple. Players toss acoin to determine whogoes first, then taketurns placing theircounters on the unoc-cupied cells of the board.A player wins by con-structing a chain of coun-ters joining the two edgesthat he or she owns. The chainmay have many branches andloops—all that matters is that thecounters form a connected path fromone edge to the opposite edge. It lookssimple, but that appearance is deceptive.Hex is a game of deep subtlety.

The game was invented by Piet Hein, aDanish mathematician also known forhis poetry. He called the game Polygon,and it first appeared in the December 26,1942, edition of the Danish newspaperPolitiken. It was independently reinventedby mathematician John Nash in 1948,when he was a graduate student at Prince-ton University. Nash later won the NobelPrize in Economics for his work in game

theory. At Princeton the game was knownas Nash, or sometimes John, because itwas often played on hexagonal bathroomtiles. In 1957 Martin Gardner wrote aboutHex in this column, and it became anovernight craze in virtually every mathe-matics department in the world.

Some simple analysis can illuminatethe game. The number of moves isfinite—a maximum of 121 for the 11-by-11 board—and a connected chain fromedge to edge for one player necessarily

blocks any connected chain for the otherplayer. This makes it intuitive that even-tually one player or the other must win: aplayer can be prevented from forming awinning chain only if the other playercreates such a chain first. It can also beproved that with optimal play, the firstplayer should always win.

The proof uses a technique called strat-egy stealing. Suppose, for the sake of ar-gument, that the red player goes first andthat there is a strategy that guarantees awin for blue, the second player. If so,then red can figure out what that win-

ning strategy is and use it to defeatblue. Assume that red, after placing

her first counter on the board,promptly forgets her opening

move. She then pretendsthat blue is opening the

game and that she is thesecond player, not thefirst. Whatever moveblue makes, red playsthe correct responseaccording to the al-leged second-playerstrategy. Sometimesthat strategy will re-quire her to place acounter in the cell oc-

cupied by her “forgot-ten” opening move, but

this poses no problem: be-cause one of red’s counters

already occupies the desiredcell, she is complying with the

strategy. She therefore makes a newmove, in any unoccupied cell, and this

becomes the new “forgotten” move.Continuing in this way, red can force a

win. But now we find ourselves in a curi-ous situation. By stealing the alleged sec-ond-player strategy in this manner, redhas played first and won, no matter whatmoves blue makes. The only way out ofthis logical impasse is to conclude that nowinning strategy exists for the secondplayer. And because the game is finite andone player must eventually win, a win-ning strategy must exist for the first player.

At first sight, this proof renders the

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Hex Marks the SpotIan Stewart shows how to make some winning connections

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BALL GAME of a different sort

can be played on a sphere tiled

with hexagons and pentagons.

(The sphere’s 12 pentagons lie

along the rim of this figure, so

they cannot be seen from this an-

gle.) The first player to surround a

cell (either unoccupied or occu-

pied by the opposing player) wins

this version of the game of Hex.

Copyright 2000 Scientific American, Inc.

game pointless, because both playersknow who ought to win if they exerciseperfect play. But the proof does not tell uswhat the first player’s winning strategy is.In fact, the largest board for which a win-ning strategy is actually known is 7 by 7!So even on an 8-by-8 board, the first play-er knows that in principle she ought towin but has no idea how to go about do-ing it. And if that still doesn’t seem fair tothe second player, many people allow anoptional rule: after the first player makesthe opening move, the second playermay elect to swap this counter for one ofher own instead of playing on a new cell.

A full discussion of Hex would occupyabout five years’ worth of columns, so I’llfocus on just two features. The first,which rapidly becomes clear to anyonewho tries the game, is that cells do nothave to be occupied to play a strategicrole. Illustration A above shows a bridgein which two nonadjacent cells occupiedby blue are separated by two intermedi-ate cells that touch both blue cells. Aslong as the intermediate cells remain un-occupied by red, the two blue cells are ineffect already joined, for as soon as redplays on one of the intermediate cells,blue can play on the other. Hex playersoften try to build chains of bridges acrossthe board. A bridge is by no means invin-cible, however. A blue bridge can be de-feated if red manages to occupy one ofthe intermediate cells while simultane-ously threatening a winning move else-where. But this is usually a tricky task, soit is best to stop your opponent frombuilding too many bridges.

A useful general principle is that a play-er’s chain is only as strong as its weakestlink. If your opponent can attack somepart of your incipient chain with good

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BASIC HEX STRATEGY is to build bridges

from one edge of the board to the op-

posite edge. A player builds a bridge

(A) by placing counters in two nonadja-

cent cells that are separated by two in-

termediate cells (hatched ) that touch

both occupied cells. More advanced

strategies involve constructing ladders

(B). In this example, the red player

wins the ladder exchange. Last, two

problems are posed as a challenge to

readers (C and D). In each game, what

cell can red play to guarantee a victory?

Copyright 2000 Scientific American, Inc.

hopes of success, then you should try tostrengthen your own weakest link or at-tack your opponent’s. To conceal your in-tentions, it is often advisable to sneak upon your opponent’s weak points fromsome distance away.

More advanced strategies involve theconstruction of ladders, which arise whenone player tries to form a connection toan edge. Illustration B on page 101 showsthe start of a ladder, with blue to move.Blue has no option except to play at cellp; otherwise, red can force a win. By thesame token, red then has to play at cell q.If blue keeps trying to force a connectionto the same edge (and for several movesshe must do this or lose), then red isforced to keep blocking, and two parallelchains of red and blue counters extendalong the edge. What blue has failed tonotice, however, is that if this processcontinues, red will win. It is important toanticipate the occurrence of ladders andto block your opponent’s ladders beforehe or she gets started. If blue had placedone of her counters near the edge earlierin the game, she would have won the lad-der exchange.

In addition to exploring these issues,Hex Strategy discusses a host of variationson the basic Hex game. For example, the

game of Y is played on a triangular board,and a player wins by forming a chainthat touches all three edges. As with basicHex, no winning strategies for Y areknown except for very small boards. Hexcan also be played on a map of the U.S.—just use the states as cells and the north-south and east-west boundaries as theedges to be connected. In this game thefirst player, who owns the north andsouth boundaries, can force a win byplaying in California. Hex can even beplayed on a sphere tiled with hexagonsand pentagons [see illustration on page

100]. The first player to surround a cell(unoccupied or occupied by her oppo-nent) wins.

Finally, I’ve provided two Hex prob-lems for readers to ponder at their leisure.The first, shown in Illustration C on page101, is taken from Hein’s original articleon the game; the challenge is to find theone cell that red can play to guarantee avictory. If that one is too easy, try thepuzzle devised by computer programmerBert Enderton of Pittsburgh for a 6-by-6board [see illustration D]. Again, find thecell that red can play to ensure a win.

The Feedback section in this past May’s column presented an elegant sur-face with one edge and one side (below), which was sent to me by JosiahManning of Aurora, Mo. I asked readers whether this surface is topologi-

cally equivalent to a Möbius band. André Gramain of theUniversity of Tours in France responded that the prob-lem is discussed in his book Topology of Surfaces (BCS As-sociates, Moscow, Idaho, 1984). He says Manning’ssurface is topologically equivalent to a Klein bottle witha hole in it, whereas the Möbius band is topologicallyequivalent to a projective plane with a hole in it.And because the Klein bottle and the projectiveplane are not equivalent, the two surfaces must betopologically distinct. —I.S.

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