The goal is making a 3x3 tiles in the center of the world into a different pattern. Nothing will be already solved.
You can warp back and forth from where you are and the centerpoint easily so you can work on this project. Making it a fixed location helps constrain your goals as a player.
If you have some fixed goal for unlocking a tool you can make your own fixed warpable box for UX purposes. You do still have to source the right location for it.
Tools
Tools are subdivided into those that affect the bg, those that affect the xcolor, and those that affect the number. For simplicity, these types of tools do different things:
bg tools involve spreading a bg out into adjacent tiles or bridging bgs, etc.
xcolor tools involve moving xcolors around, swapping them, making them jump, etc.
number tools are mathematical. They'll typically use numbers around them and whatever their conditions are will be reflected in hue or numeric shifts made to the used number so it can't easily be reused (but the shift might be useful in other ways).
I'm not sure how you cycle through lists of tools -- my current idea involves a series of drop-down boxes but this is probably bad for UX -- lists of selectable links make more sense, with hover context. Things that have been unlocked can be assigned a keyboard key, and there should be a list of "equipped tools" somewhere.
The world is decently big, but it isn't infinite. You have to work with what you have. Its area is a multiple of screens so you can make screen jumps in a saner way. There might be more in-depth mapping so you can see more of the world at once and hunt for patterns.
Tools can have conditions applied to their usage. Basic conditions affect the tool itself, like "bridge the bg of two vertical tiles" can apply only to red tiles. A deeper level here requires specific matching fg and/or number as well.
Each condition makes the complexity of the tool go up by 1, which makes the complexity of its matching pattern go down by one.
So for example, with the "bridge the bg of two vertical tiles" tool, there are three variations:
Targets a specific color (red for example) -- requires a specific 3x3 pattern to unlock. The player position is in the center.
Targets red and requires specific matching xcolors (blue for example) -- requires a 2x2 pattern to unlock. This pattern runs southeast from the player position.
Targets red, matching blue xcolors and the number 3 -- requires a specific tile to unlock. These are very easy to unlock.