3

Projex is like Hex, in that it is connection based. It is played on a board that looks like the following

     h i A B
    g . . . C
   f . . . . D
  e . . . . . E
   d . . . . F
    c . . . G
     b a I H

That is, opposite hexes are labelled with the same letter. Players alternate putting X's and O's. If you play on a letter, you also put a piece on the opposite case letter.

The main way to win is by connecting an uppercase letter to its corresponding lowercase letter (for example, F and f).

     h i A B
    g . . . C
   X . . . . D
  e X . X X . E
   d X X . X X
    c . . . G
     b a I H

You can also connect n letters to n corresponding opposite-case letters (the above way to win is a special case, actually), where n is odd. For example, here is a connection involving, A, d, and F.

     h i O B
    g . O . C
   O O . O O O
  e . O . . . E
   O O . O O O
    c . O . G
     b O I H

This game has the property that for any board configuration without empty spaces, exactly one player is in a winning position (like in Hex).

Quoting Bill Taylor

Of course the first player has a huge (indeed automatically winning) advantage, and this can be dealt with in various ways, which is not the main theme of this post, however.

Have the designers, fans, or someone else found a way to balance this game?

Some things I've thought of:

  • Use a bigger board. Player 1 technically still has a winning strategy, mathematically speaking, but among imperfect players, skill plays a much bigger role.
  • On the first turn, the first player makes one move. On other turns, players make two moves. So at the end of each player's turn, they are one move ahead. The problem is that this kind of changes the game into a entirely different one.
  • One thing that does not work is the pie rule, since all opening moves are basically the same. The board is actually a real projective plane. EDIT: Actually, its only topologically equivalent, not isometrically. The center is objectively better than any other tile, since there is a shortest path between any two letters that goes through it.
  • 1
    This game is solved. There is no way to "balance" it. – The Chaz 2.0 Feb 28 '18 at 3:00
  • @TheChaz2.0 what is the solution? – PyRulez Feb 28 '18 at 5:22
  • @TheChaz2.0 also, it's pspace complete, so any solution would be inefficient. – PyRulez Feb 28 '18 at 5:51
  • I don't see why the pie rule doesn't work: ​ One player places two Xs and one O on the board. ​ ​ ​ I suppose the extra piece might still be too strong, but in that case I'd think three and two rather than two and one should work. ​ ​ ​ ​ ​ ​ ​ ​ – user13741 Feb 28 '18 at 10:46
  • @RickyDemer I mean the regular pie rule. I didn't think about more complicated pie rules. – PyRulez Feb 28 '18 at 16:07

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