Tag Archives: pentominoes


I mentioned in a post from earlier this month that I thought Blokus was one of the greatest games around for young and old.  Blokus is particularly phenomenal not because of the game’s published rules, but because its simplicity and flexibility allow the creation of many games-within-a-game.  In short, Blokus encourages creativity and allows the space for that to happen.

Here are some of the ways I’ve used/played Blokus successfully with my young daughters, my high school students, and with adult friends and colleagues.  I’ve ranked my suggested extensions according to how I think young learners would be able to handle them.  Just because I list something as challenging certainly doesn’t mean it is beyond any learner.  PLAY and maybe you’ll accidentally learn something!

  • First, the game as described in the Blokus rules is just phenomenal.  If you never try any of my extensions, the game is worth it.


  • I started Blokus with my oldest daughter when she was 4 or 5 years old.  My philosophy of learning-play has been to give my children as much of the “rules” as they can understand and then let them decide how they want to play. My eldest wasn’t interested in the game I described, but she loved the brightly colored pieces and the freedom it gave her to create geometric designs.  No problem.  The limited number of pieces and shapes restricted her in a way that coloring and drawing on paper never did.  She had to plan her designs to get what she wanted.  Learning happened.
  • When she finally wanted to start playing a game, she didn’t like the competitive “block your opponent” rule, so she created a rule that you placed pieces on the board so that everyone could try to play all of their pieces.  She called it “Nice Blokus”.  We each played two colors and alternated laying down our pieces.  It can be just as (or even more) challenging trying to be nice to your opponents as it is competing against them.  (Perhaps there’s a broader lesson here!)
  • When we did start playing competitively, it was interesting to help her learn to strategize.  Don’t just play, talk about better ways to play.  What pieces do you want to play first and which do you want to save for late in the game?  She doesn’t always want to talk about strategies, sometimes she simply wants to play–so I adapt.  When she does want to talk, we definitely engage in serious, thoughtful, directed, and intentional play.


  • When you glance at the various pieces listed in the Blokus game, I bet many see not much more than different shapes, but there’s some very nice geometry involved.  How many shapes can you make from n squares, all connected by their edges, if you were permitted only 1 square?  2 squares?  3?  4?  5?  This could be an early exposure to the logic of mathematics of multiple cases.
    1. Obviously there is only one way to have one square.
    2. There is also only one way to have two squares.  While the orientation may change, ultimately all connections of two squares are the same.
    3. Three squares?  I don’t think it’s that hard to figure out that there are just 2 ways to do this:  three in a row–or–in an “L” shape.  One way to establish this would be to take a 2-square Blokus piece and a 1-square piece and try arranging them in different patterns until you are satisfied you have all the arrangements.
    4. What about 4-square arrangements?  Again you could use the 1-, 2-, and 3-square Blokus pieces to discover this answer on your own.  I’ll hold this answer to the end of this post in case you want to explore.
    5. What about 5-square arrangements?  These are called pentominoes, and there are many cool extensions using these. There are 12 unique pentominoes–a factoid I must give away now because it leads to many other games.  Whether or not you tell someone there are 12, can you prove that there are only 12?

NOTE:  Each color in Blokus contains every single possible 1-, 2-, 3-, 4-square, and pentomino arrangement.  Cool.

A math colleague, LG, once described the goal of the math problem-solving process this way:  FIRST, show your answer works.  SECOND, show no other answers work My experiences with most mathematics resources and teaching is that students always deal with LG’s first criterion, but rarely deal with the necessity of the second.  The problem of determining the number of unique pieces you can make with a given number of squares requires careful logic to make sure that one has found all solutions and much more importantly, that none are missing.


  • Because there are 12 unique pentominoes, obviously their combined area is 60 units.  If you create an 8×8 square grid and exclude any 2×2 square grid from its interior, it is always possible to fill in the remaining 60 units using the 12 pentominoes.  The short video below shows a solution to one of these problems.
  • While it seems like there might be many of these “omit a 2×2 square from an 8×8 square and fill in the rest with 12 pentominoes” puzzles, in reality, there are surprisingly few.  Using rotational and reflection symmetries, I’m convinced there are only 10 unique such puzzles.  Can you or one of your learners prove this?
  • Given that there are only 10 of these fill-in puzzles, how many unique solutions are there to each?  I don’t know the answer to this question.
  • In researching for this post, I encountered another interesting puzzle (link in the next bullet). In short, the total area of the pentominoes, 60, is equivalent to 6×10, 5×12, 4×15, and 3×20.  Use the 12 unique pentominoes to fill a rectangle defined by each of these four dimensions.  Even though 60=2×30, explain why it isn’t worth trying to use pentominoes to fill a 2×30 rectangle.
  • The last puzzle is nice, but going back to LG’s admonition, how do you know you’ve found all possible solutions?  This is a very hard problem to do and likely requires a computer search according to the Tiling Rectangles section in this article.


I’m hoping these examples have convinced you that Blokus has the potential to be so much more than just a box game.  Be creative.  Explore.  Maybe you’ll accidentally learn something.  Most of all, HAVE FUN!!!


This next image shows the singleton solutions for the 1- and 2-square arrangements (yellow) and the two 3-square (green) arrangements described above.  There are five 3-square arrangements.  There are several ways to create convincing logical arguments that all arrangements have been found.  My personal approach was to imagine arrangements (from left to right in the red) with all four in a row, then with three in a row, and finally with no more than two in a row.

I mentioned earlier that there are 12 pentominoes.  The image below shows them all.  How can you extend my logic from the 4-square pieces to account for all of the pentominoes?

All of this makes me wonder if there is a numerical pattern lurking here.  If there is 1 1-square piece, 1 2-square piece, 2 3-square pieces, 5 4-square pieces, and 12 5-square pieces, how many n-square arrangements would there be?  Right now, I don’t know the answer.