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watercolor or any type of water-based paint doesn't work/ watercolor or any type of water-based paint don't work

Is it doesn't or don't? Plural or singular?

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The use of or means that you treat the subjects individually.

Cake or pie sounds good. [Two singular subjects.]
Cake and pie sound good. [One combined plural subject.]

In your example sentence, you're lucky that you are also using any, because any takes a singular form:

1.(Watercolor) doesn't work.
2.(Any type of watercolor) doesn't work.

Therefore, you can combine the two singular subjects with the or conjunction and still use a singular verb:

Watercolor or any type of water-based paint doesn't work.


But had you used all types, for instance, you'd be in trouble:

1.(Watercolor) doesn't work.
2.(All types of watercolor) don't work.

In this case, the first subject is singular, so it uses a singular verb. However, the second subject is plural, so it uses a plural verb. Whichever verb form you chose with the or conjunction would fail to be grammatical.

Watercolor or all types of water-based paint doesn't work.
→ ✘ All types of watercolor doesn't work.

Watercolor or all types of water-based paint don't work.
→ ✘ Watercolour don't work.

The only way to get the combined sentence to work with all types would be to replace or with and, turning it into a combined plural subject:

Watercolor and all types of water-based paint don't work.


Incidentally, the same is true if you turn watercolor into watercolors, making the first subject plural, but leaving the second subject singular:

Watercolors or any type of water-based paint don't work.
→ ✘ Any type of water-based paint don't work.

As before, when there are mixed singular and plural subjects, it can only be fixed by using and rather than or, making it a combined plural subject:

Watercolors and any type of water-based paint don't work.

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