Echoing what other people are saying, even if you accept the premise that you want to add multiple colors, adding 10 is excessive. Beyond the fact that it would be undraftable, exponentially increase the number of 2 color archetypes that are possible, create so many colors that people wouldn't be able to hold the identities of each one in their heads, you also wouldn't even have much space to explore each color. If you made a set with 15 colors, 150 commons and each of the commons was monocolor, you still only have 10 commons to completely define each color.
I think the absolute limit you could do and still have a chance of making something playable would be adding 5 colors, each one being the bridge between ally color pairs, and using ally tri lands at common (for example, if you put purple between blue and black and brown between black and red, you would have a Blue/Purple/Black land, a Purple/Black/Brown land and a Black/Brown/Red land in that slice of the color pie).
At that point I think you might both make playing a limited deck with one of 3 color mini-arcs possible (it would still require a lot of work), then you need to create an environment where the new colors have an equal number of mechanics to the old colors. This means you either have to invent mechanics that are not in the game to equal the existing ones, simply allow your new colors to share mechanics with old ones, or eliminate mechanics in old colors and move them to new colors for your environment. Personally, I think you're going to want to come up with as many new mechanics as possible because I don't get the purpose of adding colors just to make them do what the old colors already did. One thing I would do is try to come up with mechanics that have existed in specific sets/blocks in mtg but either were only on colorless cards or were given a color that didn't really fit.
I think the absolute limit you could do and still have a chance of making something playable would be adding 5 colors, each one being the bridge between ally color pairs, and using ally tri lands at common (for example, if you put purple between blue and black and brown between black and red, you would have a Blue/Purple/Black land, a Purple/Black/Brown land and a Black/Brown/Red land in that slice of the color pie).
At that point I think you might both make playing a limited deck with one of 3 color mini-arcs possible (it would still require a lot of work), then you need to create an environment where the new colors have an equal number of mechanics to the old colors. This means you either have to invent mechanics that are not in the game to equal the existing ones, simply allow your new colors to share mechanics with old ones, or eliminate mechanics in old colors and move them to new colors for your environment. Personally, I think you're going to want to come up with as many new mechanics as possible because I don't get the purpose of adding colors just to make them do what the old colors already did. One thing I would do is try to come up with mechanics that have existed in specific sets/blocks in mtg but either were only on colorless cards or were given a color that didn't really fit.
EDIT: Terrible hungover math.
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