MTH110	Additional Tilomino Notation for Assignments

Table of Contents

• Introduction and Page Guide
• Tilomino Notation
• Addtional Tile Notation
  1. Simple Notation for Unnamed Tiles
  2. Generalized Tile Notation
• Set Notation and Sets
  3. Set Notation for Tilomino Worlds
  4. Other Tilomino Sets
• Functions and Relations
  5. Additional Functions
  6. Additional Relations

Note that you will be expected to be able to answer questions on Tilomino in the midterm and final exams.

The sections on this page are in order. This means that in order to understand a specific section, you will need to read the sections that precede it.

Tilomino Notation

Simple Notation for Unnamed Tiles

If, in your assignments, you need to refer to a tile which does not have a name, you may reference it by (row, column), starting from (1,1) in the top left corner. For example, if you need to discuss the unnamed circle in the The World is Round world you can refer to it as (6,8).

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.

Generalized Tile Notation

In many of the assignment questions, you will only work on one world at a time, so simple tile notation (e.g. a, b, c, (6,8)) will be enough.

However, in some of the questions we will need to discuss tiles in more than one world. This will be problematic because the same tile names (e.g. a, b, c) are used from one world to another even though tiles of the same name in different worlds are not the same. For example even though the All Here world has a large square called a in position (8,8) and the All Here Revisited world also has a large square called a in that same position, these two tiles are not the same, simply because they are in different worlds.

Therefore, if we want to discuss a collection of tiles from different worlds, we need a more precise naming mechanism for tiles that not only names the tile that is being discussed but also the world containing this tile.

The notation used is:

[world:tile] which refers to a specific tile in a specific world.

For example:

[R:a] is tile a in world R,
[S:(7,4)] refers to the tile on cell (7,4) in the Smiling World.

For the earlier example above, [A:a] refers to tile a in the All Here world, whereas [H:a] refers to tile a in the All Here Revisited world.

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.

For a general introduction to the set theory concepts needed to understand this section, please read section 5.1 of the Epp textbook pages 255 to 262.

Set Notation for Tilomino Worlds

A Tilomino world is really a set of tiles. Therefore we will use the name of a world to refer to the set of tiles in that world.
For example, 2 = Two By Two = {a, b, c, (8,4)}

Of course, worlds can also be defined using the generalized tile notation: For example, 2 = Two By Two = { [2:a], [2:b], [2:c], [2:(8,4)] }

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.

Other Tilomino Sets

In addition to the Tilomino worlds just described, some of the assignment questions will refer to the following sets:

• TILOMINOUNIVERSE = {all possible Tilomino worlds}
  = {all possible 8X8 grids where some of the cells contain a Tilomino tile}
• WORLDS = {all worlds in the Tilomino program} = {F, 2, R, A, B, S, H}
• TILENAMES = {names of tiles in Tilomino} = {a, b, c, d, e, f, g, h, i, j}
• SHAPES = {shapes of tiles in Tilomino} = {circle, square, triangle}
• SIZES = {sizes of tiles in Tilomino} = {small, medium, large}
• LABELS = {labels of rows and columns of Tilomino} = {1, 2, 3, 4, 5, 6, 7, 8}
• TILOMINO = {x w | w WORLDS}
• Given any i LABELS, and any w TILOMINOUNIVERSE, define
  • ROW(i,w) = {x w | x is in the i^th row of world w}
  • COL(i,w) = {x w | x is in the i^th column of world w}
  
  
• Given any i in LABELS, define
  • ROWS(i) = {row(i,w) | w WORLDS}
  • COLS(i) = {column(i,w) | w WORLDS}

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.

Additional Functions

The assignments may also refer to the following functions.

Given any tile x and any w TILOMINOUNIVERSE, define

• ROWOF(x,w) =
  • 0 if x w;
  • n LABELS s.t. x ROW(n,w) otherwise;
• COLOF(x,w) =
  • 0 if x w;
  • n LABELS s.t. x COL(n,w) otherwise;

For example: ROWOF(e,B)=7 and COLOF(e,B)=3 but ROWOF(e,2)=0 and COLOF(e,2)=0 because there is no e tile in world 2.

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.

Additional Relations

The assignments may also refer to the following relations:
• In the set TILENAMES, the < relation is the alphabetical order. (i.e. a < b < ... < j)

  If x < y we say that x has a lower name than y and that y has a higher name than x.

Remember: this notation is only for your assignment description and writeup. It will not work in Tilomino.