How do we write modular notation? I have searched but other answers do not satisfy. Yes N = 1 (mod 2) means N mod 2 is 1.
But how do I write ((N mod (K +1)) - 1) in a formal paper? I want to write steps and use a variable. For example: To win, use P numbers in the game. Where, P = ((N mod (K +1)) - 1)
On second thought - there are two different things that get called "mod".
One is an equivalence relation; grammatically, we treat it as an adverb modifying the equals sign. In this form, no further arithmetic "outside" the mod is allowed, because in principle it applies to the whole equation. If you are using this, what you wrote is formally incorrect.
The other is an operation, the remainder after integer division. It's common in programming, often with the % sign in code. In this case, the remainder $(N\mod K+1)$ will specifically be between $0$ and $K$, the representative of the equivalence class in that range. Well, usually. Some programming languages will flip this to the range from $-K$ to $0$ for negative $N$.
I should also note that your specific example is problematic here, too - what happens if that remainder is zero? How can you use $-1$ numbers in the game?
If you're going to use that operation, you need to clarify what you're doing, in words. See here for more on that subject.
