How to play
BAND
SETUPYou will need . . . "Elements"

Set out a collection of colors.

The game will involve manipulating bands of these colors.

The relative sizes of continguous blocks of color aren't important -- the bands in this game are stretchy.

"Equalities"

Decide on some bands to call equal (or interchangeable).

Let's allow these to be swapped out for one another.

And let's allow these all to be swapped out for one another.

The colors and equalities you set out are just a starting point, and as the game progresses more of them will be created.

INSTRUCTIONSSwapping

From the equalities you have, you can deduce more equalities that were not already known.

To find more bands equal to a particular band, repeatedly swap out contiguous segments of it for their equals.

Renaming

Any band can be treated as an individual.

Let's use green as shorthand for blue-yellow-red.

At any time in the game for convenience you can introduce a new color equal to a band of existing colors.

PLAY

See what equalities and pieces can be made from the ones given.

Help
  • The upper panel displays the collection of pieces available in this game.
  • The lower panel displays the equalities between bands you've discovered.
  • Drag and drop pieces to and from the center area to build a band.
  • You can then select parts of this band with your cursor.
  • Once a part is selected, anything you've already discovered equal to it will appear next to it.
  • To find new equalities, swap out equal parts of a band. (Make selections and then click on options that pop up.)
  • You can also click the pencil icon next to a selection to rename it, thereby creating a single piece to use as shorthand.

Pro tip: You can hold Control/Command or Shift to make selections with more precision.

VARIATIONS

If you completely remove equalities and renaming, so that bands of pieces can no longer be treated like individual pieces, you will just be making bands.
Note: This is not the same thing as setting up the above game with no equalities to start (a so-called "free band").

If you stop keeping track of left-to-right placement of colors, you get a semilattice.

If you allow keeping track of multiple copies of a single color in a row, you get a monoid.

No concept of order

Semilattice

Band

Monoid
No concept of multiplicity

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