The Rules of Clump

The rules of Clump are quite simple:
  1. Two players play alternately.
  2. The player moves by placing his or her symbol into an open hex (one without an 'X' or an 'O').
  3. Play continues until one player has a clump.
  4. A clump is a group of three or more connected hexes.
  5. The player who got the clump is the loser.
In this implementation, the hexes which make up a clump are shown in purple. Here's an example of a clump:

The group of four X's hexes at the bottom form a clump.

The following diagram shows the board before X made his last move. He's in trouble - playing any of the open hexes causes a clump!

If you think the definition of a clump is not particularly precise, well, you have a point. Just what does it mean to say that a clump is a set of three or more "connected" hexes? Connected means what?

Here's a more formal definition:

Let x represent the minimum distance between any two pairs of distinct hexes. Two hexes are "adjacent", if the distance between their centers is equal to x. A set of hexes forms a "block", if for any two hexes H1 and H2 in the set, there is a sequence of hexes in the set which starts with H1, ends with H2, and has the property that any pair of consecutive hexes in the sequence are adjacent. An "assigned block" is a block in which each hex contains a symbol (either "X" or "O"), and the symbol is the same for all hexes in the block. A clump is an assigned block which contains at least three hexes.

Does that clear up any lingering questions?

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