# Why "++++-++++++" is a win.

• Update:

Thanks bravejia for pointing out one case which was missing in my original post.

the winning strategy is that after flip, two strings are still both can-win-or-lose strings. Basically, starting player can choose not to make decision in his first turn.

Take "++++-++++++" as an example again:

A: ++++---++++

This will leave B with two both can-win-or-lose strings. And there will be no further subset can-win-or-lose strings.

Orignal Post:

I think "++++-++++++" is a lose. And here is why.

Both ++++ and ++++++ are can-win-or-lose strings. A can-win-or-lose string is a case where first player can choose if he wants to win or lose.
For example, in case of ++++, first player can choose flip middle two to win, or to flip first two to lose.

So in case of two can-win-or-lose strings, or even number of can-win-or-lose strings, second player can always adjust his strategy based on first player's move.

Take "++++-++++++" as an example,

If A choose to win "++++", B can win by winning"++++++".

A: +--+-++++++

B: +--+-++--++

If A choose to lose "++++", B can win by losing"++++++".

A: --++-++++++

B: --++-+--+++

B can have same strategy If A choose to flip second can-win-or-lose strings first.

So "++++-++++++" is a lose. Feel free to correct me if I am wrong.

• If I was the starting player, I'll flip this ++++-++++++ into this ++++---++++. I'll win no matter what. Let me know if this make sense.

• I see. I missed this case where after flip, two strings are still both can-win-or-lose strings. Basically, starting player can choose not to make decision in this case. Thanks for your input.

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