Thursday, May 5, 2016

Blokus: A strategic game of spatial thinking

Last night, I played Blokus (Amazon affiliate link here) with a few friends (none of which consider themselves to be math people).  In Blokus, four players take turns placing one of their colored tiles on the board, starting at their corner. Each piece played must touch another piece of the same color, but only on the corners.  As the board fills up, players must block their opponents, protect their territory, and plan out a strategy in order to fit as many of their remaining pieces as possible.  The game requires a ton of spatial thinking, and can be enjoyed by everyone from early elementary schoolers to adults.

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My roommate asked an awesome question after we had finished our game: If we worked together, could we fit all the pieces on the board and still follow the rules of starting from a corner and having same-color pieces touch by corners? I immediately started unpacking the pieces so that we could give it a try.  Before getting started, we did a few things:

1. We counted how many squares there were on the board and on all of our tiles, so that we could get a sense of how tightly we would need to pack them in.  The board was 20 by 20, and hence had 400 squares.  Each color had 1*1 + 2*1 + 3*2 + 4*5 + 5*12 = 89 squares, so there were 89*4 = 90*4 - 4 = 360 - 4 = 356 squares on the tiles.  This means that we can leave 44 squares uncovered.  Note that all of this was talked through out loud without needing paper or a calculator.

2. We decided that we would allow ourselves to move around pieces after placing them.  We just wanted our final product to be a valid Blokus arrangement, and so didn't care about following the usual rules during the placement process.

3. My roommate suggested that we start by mirroring each other, as we had often done that for the first few moves while playing the game.

My neighbor took the lead most of the time in placing pieces, and we quickly realized that copying someone else's placement really challenged our spatial thinking skills, especially as the colors began to mingle.  We liked the symmetric arrangement, and decided to try and keep that going the entire time.  We also realized that it was easier to fill in the corners as much as possible with two colors, and then worry about connecting each color later:

A recreation of our board after filling in the corners

Here was our final result, which we were all very happy with:


Although there were no children involved, this is a great example of how to encourage more mathematical thinking at home! If my roommate's question had been shot down, we would have just put away the game and been done for the night, but we followed the principal of "Yes, and..." and created something we were really proud of (as evidenced by the fact that a picture was immediately posted to Facebook).  Mathematics is about asking and exploring interesting questions, and can happen naturally in the right sort of environment!

Although this didn't come up last night, I wanted to mention another mathematical aspect of Blokus:

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. They are classified by how many square cells they contain. A monomino has one cell, a domino has two cells, a tromino has three cells, a tetromino has four cells, etc. How many different tetrominos are there? Well, it depends on how you define different.

How many different tetrominos are pictured above?  
Since Blokus pieces can be rotated and flipped, it makes sense in our situation to consider all of the ones pictured above to be the same.  It might be fun to take out a piece of paper and try to draw and count all of the polyominos with one, two, three, four, and five cells.  You will find 21 shapes which correspond exactly to the Blokus pieces!

21 Blokus pieces in red
Counting polyonimos quickly becomes a very hard problem - the number of polyonimos will n cells is only currently known up to n = 28.  They are great ways to explore area, perimeter, symmetry, and counting in the elementary school classroom and beyond!

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