Blokus in space.

The Blokus family of games, for 2 to 4 players (see www.blokus.com; Gough, 2006; Gough, in press), are some of the best spatial thinking games ever devised.

The original version of Blokus is geometrically two-dimensional, although, of course, each playing piece has physical thickness or "depth." Blokus (the first) uses all the mathematically distinct polyominoes, from monominoes (unit squares) up to pentominoes (flat shapes, made using five unit squares, joined by whole edges). Two polyominoes are not distinct, or different, if it possible to pick one of them up and turn it around, or flip it over, and place it exactly on the other one--that is, not being distinct is equivalent to being congruent, geometrically.

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Blokus Trigon (the second) is a natural development of the polyomino version, made by simply replacing unit squares with unit triangles. (Changing a mathematical situation by altering a building block is one version of problem-posing.) The result is a game that uses polyiamonds (the polyomino-like family of two-dimensional shapes, built using unit-sided equilateral triangles) to make a set consisting of all the mathematically distinct polyiamonds, from the unique moniamond to all of the hexiamonds.

In both of these 2-D versions of Blokus, each player has a set of pieces (polyominoes, or polyiamonds) of one colour. Players take turns to place one of their own coloured pieces on a board (either a square grid, or a triangle-based hexagonal grid). Players aim to use as many of their own pieces as possible, while blocking (hence the games' names) opportunities for their opponent or opponents. The rule for placing a piece is simple: the piece played must touch at least one other of that player's own pieces, but can only touch at a corner--that is, pieces of the same colour cannot touch along a shared edge. Very simple. Three-year olds can learn this.

The latest (third) version, Blokus 3D, takes these Blokus games to a whole new level. (I couldn't resist the pun, which happens to be literally, spatially, true!) Instead of using flat pieces made of unit squares or unit triangles, Blokus 3D uses mathematically distinct objects made of unit-cubes, joined by whole faces, like tetracubes, or Soma cube pieces (e.g., Gough, 2001; 2007; in press). This is another example of problem-posing by changing the basic building block. However, I doubt that we can expect further Blokus variants: we have exhausted the possible building blocks. I cannot imagine an effective space-filling game based on tetrahedrons, nor a tessellating game based on hexagons.

Note that using three-dimensional "solid" objects allows for the possibility of two objects being exactly the same except that they are mirror images of each other. As with a right-hand glove and a left-hand glove, no simple picking up, turning, or flipping can transform one into the other!

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These Blokus 3D materials can be used by one player as a three-dimensional jigsaw-type puzzle. (A puzzle may be described as an activity that one player can solve only by lucky random trial and error.) Otherwise, two, three or four people can play it as a competitive spatial game, rather like the classic Pentomino game, placing pieces, one after the other, to occupy the spaces of the playing board, and the "heights" above the board. This also resembles Tetris, in the way successive pieces can "slot" on top of others; although, in the case of Tetris, the "playing" is more puzzle-like: one person versus the computer's randomly generated Tetris pieces and the time-limit of falling pieces.

Since the Blokus 3D playing pieces are made from unit cubes, the playing board is a square grid based on unit squares--but with a twist. Depending on the players' choice, Blokus 3D can be played using several very different playing boards: rectangular shapes (oblongs and squares), and irregular rectilinear shapes (like a block-letter upper-case L), each "framed" by a template placed on the grid board.

Moreover, players must fit their pieces within specific building profiles, such as a cuboid, a staircase, or a ziggurat (a step-pyramid).

Perhaps surprisingly, each different frame results in slightly different strategic thinking. This is very interesting: players will aim to leave as many upper-faces of their own pieces placed so that no opponent can cover them. Uncovered faces contribute to the way a player wins, involving Nim-like tactics, making sure successive columns of a specified maximum altitude are "filled."

Each player uses a set of pieces of one colour: red, green, blue or yellow.

Each set consists of mathematically distinct polycubes, starting with the one dicube, the two possible tricubes, and the 11 distinct tetracubes.

Obviously this uses 52 unit cubes altogether.

Here is the unique dicube, one of the tricubes, and a tetracube:

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Notice that it is actually a tetromino, made of unit cubes. Although the image shows it "standing up," we can describe it as a "flat" tetromino--but not all tetrominoes are "flat:" some have a height, when sitting flat and squarely on the table, of two units.

Scoring may be optional, but the winner can score the difference between his or her total of visible upper-faces and the next smaller number of visible upper-faces of an opponent. For example, on a 5 x 4 base, red may win with 6 visible upper-faces, and green may have 4 visible upper-faces, with blue and yellow each having 5 visible faces: red scores 1 point.

Exercises

1. What are the 11 mathematically distinct tetracubes?

2. What is the maximum "footprint" of all the pieces in a player's set: that is, what is the greatest total area underneath (the number of unit-squares covered on the playing board) when all of a player's pieces are placed on a square grid board?

3. Which of the tetracubes and tricubes are also pieces of Piet Hein's famous Soma cube? (Blokus 3D is a wonderful Soma resource because it includes four differently coloured Soma sets! Great for Soma games and activities in Gough (2001).)

4. Which of the Blokus 3D pieces can be used to fill space ("filling" a volume of space with no gaps or holes--the three-dimensional equivalent of two-dimensional tessellating covering a surface) using one of the pieces repeatedly; or one pair of pieces, or one triplet of pieces, and so on, repeatedly?

5. Ignoring when a 3D Blokus piece has a maximum height greater than 1 unit, which of the pieces can be used to tessellate? Try each piece, repeatedly, on its own; try pairs of different pieces, and triplets of different pieces.

DIY tetracube 3-D Blokus game This introductory two-player variant is based on the official Blokus 3D rules and pieces. You need 44 unit cubes to make a one-colour set of the playing pieces.

One player uses a red set of the 11 distinct tetracubes. The other player uses a blue set of tetracubes. The playing board is a 5 x 5 grid of unit squares.

Players take turns to place any one of their unused pieces on the playing board. Once a piece is placed, it cannot be moved.

The first player places a piece anywhere on the board, fitting in the grid of squares. Then each new piece must touch at least one other piece. (Optional rule: after the first piece, each new piece must touch at least one face of a piece of the same colour.)

As play progresses, no face underneath the piece being placed can "float," unsupported, like the ceiling of a room. No part of a piece can extend beyond the boundaries of the chosen playing area. No piece can be placed if the resulting height exceeds four cubes.

The game ends when no further play is possible.

The winner is the player with more "roof" faces of his or her colour visible in a bird's-eye view of the board.

References and further reading

Gough, J. (2001). Learning to play: Playing to learn--Mathematics games that really teach mathematics. Brunswick: Mathematical Association of Victoria [MAV].

Gough, J. (2006). Editorial: Do you know Blokus? Vinculum, 43(4), 2.

Gough, J. (2007). Diversions: Polyominoes and polyiamonds--Blokus. The Australian Mathematics Teacher, 63(4), 16.

Gough, J. (2009). Meet Blokus in three-dimensions. Vinculum, 46(2), 18.

Gough, J. (in press). Brain-boosting mathematics games: Are you game? Adelaide: The Australian Association of Mathematics Teachers [AAMT].

with John Gough 

Deakin University 

[e-mail]

Note

Blokus 3D is a renamed development of the earlier game called "Rumis" designed by Stefan Kogel, a multi-award-winning game inventor!

RELATED ARTICLE: Variants.

* Change the height limit to a maximum of 3 cubes or 5 cubes.

* Use a 4 x 6 grid playing board, and a maximum height of 5 cubes.

Variants for a standard Blokus 3D set

* Use a 5 x 5 square grid, with players restricted to placing pieces, in the usual way, no higher than a 5 x 5 x 4 cuboid or a 5 x 5 x 3 cuboid.

* Alternatively, use a 6 x 6 square grid, with a maximum height of 4, 3 or 2.

* Use Blokus 3D pieces, and standard Blokus rules (a player's next piece must only touch one of his or her pieces already played only at a "vertex," that is, one shared vertically-aligned edge), to play on a 10 x 10 board, with no stacking (or allow stacking no higher than 2 cubes): winning as in standard Blokus, the player with fewest unplaced unit-cubes is the winner.

* Devise your own different playing board "frames"--what about a block-letter capital T-shape?

* Devise your own side-view profiles: how different is the game if the specified "stair case" profile is changed to a flat-roof design of a designated height?