On Optimal Play in the Game of Hex

Hex is a two person game played on an n×n board in which the players take turns trying to construct paths from one side of the board to the other. It is known that there exists a winning strategy for the first player, but no one has yet been able to find such a strategy for any board larger than 9×9. Despite this, we ask the following two questions: “what is the shortest path with which player one can guarantee a win?” and “what is the minimal number of moves player one must make to guarantee a win?” We give lower bounds on answers to these questions and conclude with a number of conjectures and “challenges.” 1. A Brief History of Hex In 1942, Piet Hein invented a game he called Polygon while contemplating the (then unsolved) four-color theorem. The game was later independently re-discovered by John Nash while he was a graduate student at Princeton University and it quickly became popular there (and at other schools) under the names of John and Nash. A couple of years later, in 1950, Parker Brothers began producing the game as “Game of Hex.” Figure 1. A picture of the original game marketed by Parker Brothers [1]. For copies of the original instructions see [7]. This work completed with support of the Woodrow Wilson Career Enhancement Fellowship and Swarthmore College. INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #G02 2 With so many names given to the same game, one might wonder why “Hex” is the name that endured. The fact that Parker Brothers put its marketing muscle behind that name is clearly a large contributing factor, but Martin Gardner’s 1957 Scientific American article [5], Concerning the game of Hex, which may be played on the tiles of the bathroom floor, is undoubtedly another. This article and future writings by Gardner on the game were quite popular and brought the game to the attention of a vast number of mathematicians and computer scientists. In the article, Gardner states that Hex may well become one of the most widely played and thoughtfully analyzed new mathematical games of the century. So what exactly is this game of Hex, how does one play and what makes the game so special? First, Hex is a two person game played on an n × n board of hexagonal tiles arranged in a rhombus. We will denote the game of hex played on an n× n board as Hex(n). The Parker Brothers board shown in Figure 1 is an 11 × 11 board and is considered the “standard” size board. Two smaller 4× 4 boards are illustrated below. Figure 2. An empty 4 × 4 Hex board (left) and a 4 × 4 board containing a winning path for blue (right). Each player is assigned a pair of opposite sides of the board and given a set of counters. The players take turns placing one of their counters on any unoccupied tile of the board until one player has a path consisting of their own counters connecting their two sides of the board. Throughout the discussion here, one player is designated blue, the other red and we color the sides of the board and the counters to indicate to whom they belong. We also make the arbitrary choice of always letting the blue player go first. Figure 2 above illustrates a winning game of Hex(4) for blue. Hein and Nash each proved independently that Hex cannot end in a tie and moreover, that there exists a first player winning strategy. Proofs of these facts can be found in [4], [6] and [10], among other places. (Most notably, in [4], David Gale proves that the inability of Hex to end in a tie is equivalent to the Brouwer fixed point theorem.) For small n, it is a simple matter to work out first player winning strategies for Hex(n), but as n grows, this task quickly becomes quite difficult. Jing Yang was first to find a first INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #G02 3 player winning strategy for Hex(7) [11] and seems to have recently found such strategies for Hex(8) and Hex(9) [13]. The search is still on for a winning strategy for Hex(n) when n > 9. We note that while we will mostly be concerned with Hex played on n × n boards, the game can also be played on asymmetric, m× n,m 6= n, boards as well. However, in this case, the player with the smaller side can always win and a winning strategy is quite easy to describe (see [6], for example). 2. Introduction to the problem Despite the fact that no one is able to describe winning strategies for Hex(n) when n is large, we examine two possible measures of optimal winning strategies. In particular, we ask the following two questions: Question 1 What is the shortest path with which player one can guarantee a win? Question 2 What is the minimal number of moves player one must make to guarantee a win? To address the first question, we let Hex(n, l) denote the game of Hex(n) with the added constraint that player one wins only by constructing a path of length less than or equal to l. For example, the winning path in Figure 2 would constitute a win for blue in Hex(4, 7) but not in Hex(4, 5). Furthermore, if Figure 2 represents the state of the board during a Hex(4, 5) game, play would continue since the Hex(4) winning path is not with a path of length less than or equal to 5 and since blue still has a chance of winning with such a path. However, since it is red’s move next, the game should conclude in a draw on the very next turn. This illustrates that Hex(n, l) can end in a tie for some values of n and l, but note that because player one has a winning strategy for Hex(n), the second player can never have a winning strategy for Hex(n, l). Now suppose we define λ(n) to be λ(n) = min{l | player one has a winning strategy for Hex(n, l)}. Question 1 is then equivalent to: Question 3 What is the value of λ(n)? Unfortunately, for n large, computing λ(n) is almost certainly more difficult than finding a winning strategy. Therefore, instead of trying to calculate the value of λ(n) INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #G02 4 precisely, we look to finding bounds for λ(n). This task also proves to be quite challenging. In particular, it is trivial to see that λ(n) satisfies n ≤ λ(n) ≤ dn2/2e and λ(n) = n for n < 4, but significantly improving these bounds appears to be far from trivial. In the following sections, we focus our attention on the lower bound and prove: Theorem 4 λ(n) > n for all n ≥ 4. Remark 5 We note that it was known [6] as early as 1959 that λ(4) ≤ 5 and that λ(5) ≤ 7. Together with the theorem, this implies that λ(4) = 5 and 6 ≤ λ(5) ≤ 7. It has also been known since that time that all but four opening moves for the first player result in a loss for that player in a game of Hex(4). In Section 7, we give a Hex(4) second player winning strategy for 10 of the 12 possible opening moves which admit such a strategy. A second player winning strategy for the remaining 2 opening moves is outlined in the appendix. Turning our attention to Question 2, we define the game Hexd(n) to be the game of Hex(n) with the added constraint that player one must win by playing at most d counters and define δ(n) to be δ(n) = min{d | player one can guarantee a win at Hexd(n)}. Therefore, Question 2 is equivalent to: Question 6 What is the value of δ(n)? This question is related to a more common measure called the depth of a game–the minimal number of total moves (made by both players) necessary to guarantee a win. The depth of Hex(n) is then 2δ(n)−1. As in the case for λ(n), it appears that (in general) computing δ(n) precisely is intractable. Therefore, we again focus on establishing a nontrivial lower bound. We clearly have that δ(n) ≥ λ(n), but we can in fact say a bit more. We prove: Theorem 7 For all n, δ(n) ≥ n+ bn/4c.