Sources page biographical material

4.B.2. HEX
David Fielker sent some pages from a Danish book on games, but the TP is not present in his copies, so we don't have details. This says that Hein introduced the game in a lecture to students at the Institute for Theoretical Physics (now the Niels Bohr Institute) in Copenhagen in 1942. After its appearance in Politiken, specially printed pads for playing the game were sold, and a game board was marketed in the US as Hex in 1952.

Piet Hein. Article or column in Politiken (Copenhagen) (26 Dec 1942). ??NYR, but the diagrams show a board of hexagons.

Gardner (1957) and others have related that the game was independently invented by John Nash at Princeton in 1948-1949. Gardner had considerable correspondence after his article which I have examined. The key point is that one of Niels Bohr's sons, who had known the game in Copenhagen, was a visitor at the Institute for Advanced Study at the time and showed it to friends. I concluded that it was likely that some idea of the game had permeated to Nash who had forgotten this and later recalled and extensively developed the idea, thinking it was new to him. I met Harold Kuhn in 1998, who was a student with Nash at the time and he has no doubt that Nash invented the idea. In particular, Nash started with the triangular lattice, i.e. the dual of Hein's board, for some time before realising the convenience of the hexagonal lattice. Nash came to Princeton as a graduate student in autumn 1948 and had invented the game by the spring of 1949. Kuhn says he observed Nash developing the ideas and recognising the connections with the Jordan Curve Theorem, etc. Kuhn also says that there was not much connection between students at Princeton and at the Institute and relates that von Neumann saw the game at Princeton and asked what it was, indicating that it was not well known at the Institute. In view of this, it seems most likely that Nash's invention was independent, but I know from my own experience that it can be difficult to remember the sources of one's ideas -- a casual remark about a hexagonal game could have re-emerged weeks or months later when Nash was studying games, as the idea of looking at hexagonal boards in some form, from which the game would be re-invented. Sylvester was notorious for publishing ideas which he had actually refereed or edited some years earlier, but had completely forgotten the earlier sources. In situations like Hex, we will never know exactly what happened -- even if we were present at the time, it is difficult to know what is going on in the mind of the protagonist and the protagonist himself may not know what subconscious connections his mind is making. Even if we could discover that Nash had been told something about a hexagonal game, we cannot tell how his mind dealt with this information and we cannot assume this was what inspired his work. In other words, even a time machine will not settle such historical questions -- we need something that displays the conscious and the unconscious workings of a person's mind.

Parker Brothers. Literature on Hex, 1952. ??NYS or NYR.

Claude E. Shannon. Computers and automata. Proc. Institute of Radio Engineers 41 (Oct 1953) 1234 1241. Describes his Hex machine on p. 1237.

M. Gardner. The game of Hex. SA (Jul 1957) = 1st Book, chap. 8. Description of Shannon's 8 by 7 'Hoax' machine, pp. 81 82, and its second person strategy, p. 79.

Anatole Beck, Michael N. Bleicher & Donald W. Crowe. Excursions into Mathematics. Worth Publishers, NY, 1969. Chap. 5: Games (by Beck), Section 3: The game of Hex, pp. 327-339 (with photo of Hein on p. 328). Says it has been attributed to Hein and Nash. At Yale in 1952, they played on a 14 x 14 board. Shows it is a first player win, invoking the Jordan Curve Theorem

David Gale. The game of Hex and the Brouwer fixed-point theorem. AMM 86:10 (Dec 1979) 818-827. Shows that the non-existence of ties (Hex Theorem) is equivalent to the Brouwer Fixed-Point Theorem in two and in n dimensions. Says the use of the Jordan Curve Theorem is unnecessary.

Winning Ways. 1982. Pp. 679-680 sketches the game and the strategy stealing argument which is attributed to Nash.

C. E. Shannon. Photo of his Hoax machine sent to me in 1983.

Cameron Browne. Hex Strategy: Making the Right Connections. A. K. Peters, Natick, Massachusetts, 2000.