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What do we know about the hex game ?

We know that John Nash has demonstrated that the first player (White) plays and wins on a hexboard Hn of order n (1), whatever n may be. But we don’t know:

Th general winning strategy because John Nash’s demonstration is non-constructive.

The tactical elements leading to a win according to this general stategy

A winning line of play for a hexboard Hn whatever n may be.

The shortest winning line of play (solution) for a hexboard Hn, n≥8, for exemple n=9.

A fortiori the solution of a hexboard Hn whatever n may be.

In this study we will find the answer to all these questions. In particular the answer to the last question is given by a congruence (mod. 6) and by 6 polynomials of second order.

All these analyses have been made by analysing only one move when White has to play and all the possible moves when Black has to play.

These analyses have used many position-types, some whith parameters, which often meet. Moreover, due to the double attack (multiple attacks) which have a devastating effect, it is possible to reduce considerably the length of the analyses, these double attacks and other manoeuvres which could greatly inspire players of the Hex game.

The systematical use of the solutions of all the positions shows that there exist relations of recurrence between the numbers of moves h(n) and h(n-6) of the solutions of the hexboards Hn and Hn-6 

                                                                              h(n)=u(n)+h(n-6) 

u(n) designing a number of moves depending on n. This relation of recurrence can only be established when n≥7.

It will be noticed that this study has been totally carried out without the help of a computer. The reader who migth wish to know to importance of the hand-written analyses must refer to the pagination of the written manuscript given in the annex.

Finally it will be noticed that the great number of analyses, carried out without the help of a computer, of chess game positions, opposing a bishop and a knight to a knight, has enabled us to finalize the method of analyse used in this study.

(1) We suggest calling hexboard Hn of order n, the support, having the general shape of a diamond, of nxn hexgonal squares on which the game of hex is played.

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