Early Pieces, Rubik Length and True Length

By | November 25, 2011

Okay, so you know that you need to move toward the middle of the board at the start of the game, and you know that you should be doing it with your biggest pieces.  But which pieces?  And which orientation should they be in?

As for which pieces, some are just better than others.  One convenient thing to think about is the “distance between the two furthest corners of one piece”.  This is called the “Rubik Length” of a piece, named for Rubik87 who was one of the first to use this idea.  For example, the i5 has a Rubik length of 6 – if you start at one corner, count corners as you move along the piece as far as you can go.  In the image below, see the Rubik length of a few pieces.

The I5 has Rubik length 6, the F has Rubik length 5, and the o4 has Rubik length 4.

 

An easier thing to count is the length in terms of squares (instead of corners).  Let’s call this the “true length” of the piece.  The true length of a piece is the minimum number of squares (of the board) between the two furthest squares (of the piece).  You cannot count squares along the diagonal, and normally you count squares within the piece (though this is not necessary – you are allowed to count squares outside of the piece if you want to – for example, the true length of the U is 4, not 5).  Notice that true length = rubik length – 1, so the three pieces in the above image have true length 5, 4 and 3 respectively.  One advantage of true length is that it is easier to figure out than Rubik length.  The Rubik length may seem to be affected by the direction you move, or the corners you choose – in actuality this is avoided by taking the two furthest corners, and picking the smallest distance between them, but that can be a bit of a pain.

The advantage of Rubik length is that every time you play a new piece, you get “one true length for free”, because you play corner-to-corner, instead of side-to-side, where as there are no “free gains in Rubik length” because it counts from corner to corner anyway.  This makes it faster to calculate how many pieces you will need to span a large area using Rubik length.  It is especially useful when considering versions of Blokus on boards larger than 20×20, but such considerations are outside the scope of this blog.

You might ask yourself this – what is the length of the 1 piece, and what is the length of the o4?  In Rubik length, the 1 has length 2, and the o4 has length 4, so it takes two 1 pieces to make up the distance of the o4, which makes sense because you place them corner to corner.  In true length, the 1 has length 1, and the o4 has length 3, which means that you need three 1 pieces to make the o4 (i.e. a path from the beginning to the end of the o4), so you are counting them as if you are “building a new piece”, not playing them one after the other.

It is likely that both the notions of true length and Rubik length will come up in this blog, as I will use whichever one is more convenient for whatever I am talking about.

True length and Rubik length are useful because it is convenient to know the maximum possible distance that a piece can span.  At the beginning of the game, you often want to use a piece to stretch as far as possible, so the best pieces to use at the start of the game are the ones with the biggest true length – they get you to the center as fast as possible.

You should check the true length of each piece as practice, and figure out which ones have the maximum (5).  You should also check the true length of the other pieces.  Hint: there are 6 pieces with true length 5.

Of course, using these pieces is good, but you want to use them for their maximum value.  The V has true length 5, so it is a good first piece to use, but the orientation matters greatly.  As you can see in the image below, the first V is a good example of a reasonable orientation for your first piece, but the second one is not.

If you are blue, and want to connect blue and red at the start of the game (which you should), you need 4 blue pieces, and 3 red pieces, and of these 7 pieces, 6 of the must have true length 5, and the other must have true length at least 4 – if you use pieces with smaller true length (total less than 34) you won’t be able to make a complete connection in the middle of the board.

This still leaves the question of which specific pieces, and which orientations you should use.  The specifics of this deserve their own post, but I will make a brief comment about the general idea:

The idea that should guide your decisions is this: you would like to place pieces that give you the ability to set up good blocks later in the game.  In general, you would like to avoid having long edges in the middle of the board, and definitely long L-shaped regions in the middle of the board.  For more on this, see the post: Shaping Your Zone

2 thoughts on “Early Pieces, Rubik Length and True Length

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