Imperviousness And Redundancy

Redundancy and imperviousness are among the most fundamental positional qualities in the game of Crumble. They are involved directly in virtually any attempt to win; and many battles taking place over the board have them as their ultimate object.

Redundancy

If a piece could safely be occupied by the opponent without breaking its chain, then that piece is redundant.

Imperviousness

Any piece that can't be swapped into without a significant amount of preparation is considered to be 'impervious'. Consider two pieces in the starting position.

With Black to move, it's trivial to split and then swap into the White piece.

The White piece was not impervious. But consider the pieces in this configuration:

How does Black swap into the big white piece now? There's no way to do it. The white piece is impervious.

Of course, Black could eventually change that situation, and swap into the white piece, if Black had enough time to do that. But typically that time is not available, or else it must be prepared gradually as a secondary aspect of attacks launched elsewhere on the board.

Imperviousness is a crucial part of winning the game. To win, you need to have a chain of pieces stretching to all four edges of the game board. If that chain is easy to interfere with, you'll never make it. But if it's composed of pieces that can't be occupied by the opponent, they become much more difficult to counter.

It's clear that to make any given piece impervious, it must be accomplished on one side of the piece at a time. So we might say a piece is 'partially impervious', or 'fully impervious', or 'impervious on one side' and so on. Typically it takes four complete turns before you can create a piece that is impervious on all four sides - one turn per side. And of course, your opponent won't be just sitting around waiting for you to gain that advantage.

An impervious piece is useless on its own. It needs to be part of a chain of same-colored pieces in order to have any value.

In the above diagram, the big white square is called "weak impervious". It's not part of a chain, so it can't participate in winning the game or performing captures. If White tries to change that by occupying one of the surrounding rectangles, it only opens up the White square to being occupied by Black.

It's true that Black will find it difficult to hold onto the big square in this configuration, but White will as well.

Now consider the better situation: a relatively strong impervious piece will have 'spikes' of pieces coming off of it, that can be used to form chains.

In the above position, the big square is called "strong impervious". It has three white spikes coming off of it, perfect for constructing chains. Even if black could occupy the big piece, the black rectangle beneath it would make that piece always vulnerable to re-occupation by White. To be considered "strong impervious", a piece must connect at least two spikes of same-colored pieces. If an impervious piece has just a single spike, it's "weak impervious", or you could just call it a "dead bulb".

Since the chain of white pieces can't lead through the impervious piece, its imperviousness is virtually useless.

Although they don't look anything alike, redundancy and imperviousness are the same kind of strength. If you have a chain of pieces threatening to win, and every piece along that chain is either impervious or redundant, how can your opponent stop you from winning?