The quilt corner, used throughout
Look at the bottom-right corner of the Quiltoku logo. Nine squares in three colours: sage, navy and burgundy. It is a complete, correct puzzle hiding inside a quilt, and Quiltoku keeps coming back to it, from the logo to the avatars you meet in friends groups.
The corner
In that corner, every colour appears once in every row and once in every column, among the big squares and among the small ones. And no big colour is ever paired with the same small colour twice. That is a complete 3×3 Double puzzle, the structure mathematicians call a Graeco-Latin square of order three. Of all the blocks you could cut out of the ten-by-ten logo, it is the only smaller one that is complete in both layers. You can read about the whole logo on Inside the logo, and about where the idea comes from in Latin boards and Euler.
Your avatar is a quilt corner
In friends groups, you do not get a name or a photo. You get a small quilt of the same kind: three colours that you choose, in both the big squares and the small ones, in a pattern the game generates. There are 72 valid patterns for any three colours, and 120 ways to choose three colours from the quilt's ten, which makes 8,640 different avatars. Each one is a valid 3×3 Double puzzle solution, so it can be read as a miniature puzzle.
The structure is also why avatars are safe. A colour can appear only once in each row and each column, so an avatar can never spell a letter, draw a cross or form a flag. It is made of the same rule you play by.
One word to go with it
Next to the quilt there is a single word: Cedar, Pinwheel, Lantern, Thistle. It comes from a list of 207 gentle words (plants, weather, places, fabrics, shapes and a few animals), each with a French partner, chosen so that none can sound rude or unkind in English or French. A word is never joined to another word, so no combination can go wrong. Inside a group, no two people share a word or an avatar, so there are about 1.8 million possible identities and only twenty people to tell apart.
A badge for the group
A group gets its own badge, also a 3×3 Double in three colours that the game picks, and one word of its own. A group of friends looks like a little patch on a wall: each member a nine-square quilt.
Why a quilt?
The game began with a quilt, City Limits, made in 1985 by Anne de la Mauvinière Silva: ten colours, squares inside squares, each colour once in every row and column twice over. That arrangement is exactly a Graeco-Latin square of order ten, a kind of square that Euler believed could not exist (the story is on the Latin page). Everything that identifies you in a group comes from the same corner of the same quilt: its colours, its structure, its rule.
Bigger squares for bigger crowds
Later, a board for many players could show everyone else only as a larger quilt, 4×4 or 5×5, with no names, and yours outlined and bold. A 4×4 Double has billions of distinct patterns and a 5×5 many more, so every player could have a quilt of their own. That is an idea, not a feature yet.