Question about winning and loosing positions at "Grundy's game"

Question

As a continue to this question: How can I know if a pile in "Grundy's game" have a winning strategy?:

I'm talking about "Grundy's game".

I understand from the answer that if I have one pile of $20$ is loosing position because $n(20)=0$,
but if I split it to $19,1$ , I get: $n(19)+n(1) = 3>0$ - and this winning position.

So my question is: how I can know if I have winning strategy or not? Assume that is my turn now and I have one pile of $20$, I can win or not?
If yes, what is me next move?

Thank you!

Answer 1

I understand from the answer that if I have one pile of 20 is loosing position because $n$(20)=0, but if I split it to 19,1 , I get: $n(19)+n(1)=3>0$ - and this winning position.

It is important to understand what "winning position" and "losing position" mean. These terms are relative to the player to move, not relative to yourself.

"Winning position" means that the player to move can win.

"Losing position" means that the player to move will lose.

If it is your move on a pile of 20, and you split 20 into 19,1, it is a winning position (as you calculated), but it is now your opponent's move, which means your opponent is the one winning, not you.

Answer 2

I understand from the answer that if I have one pile of $20$ is loosing position because $n(20)=0$,

This is correct.

but if I split it to $19,1$ , I get: $n(19)+n(1)=3>0$ - and this winning position.

This is also correct. The two statements are not in contradiction. $[19,1]$ is a winning position that you are handing to your opponent. This means your opponent can win, so you lose. A position is "winning" if the person whose turn it is has a winning strategy. Being in a winning position is good, but moving to a winning position is bad.