Is "Pour water into every a certain numbers of glasses " acceptable?

Question

There are 100 glasses arranged in a row.

When I would like someone to pour water into the 3rd glass, the 6th glass, ..., I think I can say

Please pour water into every three glasses or every 3rd glass.

I am trying to generalize this without referring to a specific number (e.g., 3rd ). My example is as follows:

1. Pour water into every a certain numbers of glasses.

2. Pour water into every Nth glass, where N is a positive integer.

I would like to allow water to be poured into every glass. Could you advise me on which is better?

Answer 1

With a specific number, like three, the usual formulation is to use an ordinal number, like this:

Pour water into every third glass.

You don't need an article. "Every third" performs the function of an article here.

If you say "Pour water into the third glass," that means that the other person should look at the first glass, skip it, then the second glass, skip it, and then pour water into the next glass. When instead you say every third, that tells your listener to repeat the same process over and over, continuing right after the previous group of three, until the last glass.

If you want to talk abstractly, using a variable like n instead of a specific number, then you have to reach a little outside common usage and borrow an expression from mathematics:

Pour water into every n^th glass.

Your second guess was right.

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See also these Google Books searches: "every third glass", "every third". Not all of the results are relevant, but looking at the relevant ones in real contexts will give you a feel for how people use this wording.