Apart from squares, there are only two other regular shapes that tesselate perfectly to form a grid: equilateral triangles and regular hexagons. So what would happen if I remade scrabble to be played with hexagonal pieces instead of square tiles?
The pieces:
Scrabble has 100 tiles, which I remade as hexagons for this version. While I was at it, I cut pieces for hexagonal Bananagrams, which has 44 more tiles than Scrabble and is played without a board.
The board:
In order to fit the hexagonal shape of the new pieces, I had to make the board a hexagon. My first design was unplayable with the bonus spaces, because I didn't account for the different way the corners of the spaces line up.
The first game is shown to the right. It used a modified version of the board with a more playable setup for bonus spaces. The arrows on the sides of the left image show the three directions of play.
An example of the two-letter rule: At the bottom left of the image that shows our first game, the word "fazed" is valid even though the word "fa" isn't valid. The player gets no points for "fa" but receives points for fazed. At the middle left, with the word "key," the word "be" is a valid word and so the player gets points for both "key" and "be."
This new game mode creates a lot of problems. Adding two more sides to the tiles means adding one more direction of play. In regular Scrabble, players place tiles left to right or top to bottom. In this game, players place tiles top to bottom at a 60 degree angle from vertical, left to right, and top to bottom at a -60 degree angle from vertical. Not only is this confusing, but it also makes placing tiles down legally according to the rules of regular Scrabble very difficult. To get around this issue, we ignored two-letter words unless they happened to be real words. This both allowed lower-level players (like me) to play words without too much difficulty and higher-level players to get a lot of points by making grids of many two-letter words. The crucial detail about this rule is that words must still have at least one valid word connection, which keeps people from placing words next to each other that don't share any letters but create a lot of two-letter pairs that aren't words.
Hexagonal Bananagrams, ignoring two-letter words completely, worked surprisingly well. The third direction of play adds an extra challenge but not in a way that makes it unplayable.
An example of the 2-letter rule for Bananagrams: on the bottom right, the words "jowl" and "prove" can be side-by-side because they both attach to the word "tile" even though none of the two letter words formed between them are valid