Best Answer:
Here is my interpretation of the question:

There are n objects in the pile.

Some number p is chosen and each player in turn

can remove any number of objects from 1 up to and including p.

And it is decided whether taking the last piece is a win or a loss.

Then someone decides who goes first and you play the game under those conditions.

Let's say p = 3 and you can remove 1, 2, or 3 objects from the pile.

The rest of the analysis generalizes to any p quite readily.

Just substitute (p+1) for 4.

Let's consider all four cases:

1. n is a multiple of 4, objective is to take the last one

2. n is a multiple of 4, objective is to not take the last one.

3. n is not multiple of 4, take the last

4. n is not multiple of 4, not take the last.

1. Multiple of 4, objective is to take the last one.

This is a losing position:

Player 1 takes some number of objects (1,2,3),

player 2 takes (4 - that number), again leaving a multiple of 4.

That continues until there are 4 left.

Player 1 takes (1,2, or 3) and player 2 takes the rest, and wins.

Player 2 wins.

2. Multiple of 4, objective is to not take the last one.

This is a winning position:

Player 1 takes 3, leaving a multiple of 4 + 1.

Player 2 takes some, and Player 1 follows the strategy above,

always leaving a multiple of 4 + 1 for Player 2.

Eventually Player 2 is left with 5, and player 1 can leave the last one for him.

Player 1 wins.

3. Not multiple of 4, take the last

Player 1 leaves Player 2 with a multiple of 4,

and now we are in the same scenario as #1, but with the players reversed.

Player 1 wins.

4. Not multiple of 4, not take the last.

Now it depends on what the actual remainder is.

If the number is a multiple of 4 + 1, it's a losing position.

No matter what Player 1 does, Player 2 does the complementary move totaling 4,

and again leaves him with a multiple of 4 + 1.

Eventually it gets down to 5, and Player 1 takes some,

Player 2 leaves 1, and Player 2 wins.

If the number is a multiple of 4+2, Player 1 takes 1,

and leaves player 2 with a multiple of 4 + 1, and Player 1 wins,

as just described.

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