Three piles of chips are to be used in a game played by Anita and Brinda: pile…

Concept

This is a subtraction game played under misere rules: whichever player is forced to remove the very last chip loses, and each turn a player may take exactly one chip from any pile or empty an entire pile in one move. Whenever only piles of size one remain, a turn simply removes one whole pile (taking one chip and taking all the chips from a size-one pile are the same move), so the two players just alternate removing single-chip piles until none remain -- the player to act with an ODD number of such single-chip piles left is fixed to make the final capture and lose, while an EVEN number lets the player to act pass that losing parity to the opponent.

Application

• Anita's opening move is to take one chip from Pile II, changing (1, 2, 3) into (1, 1, 3).

• If Brinda empties Pile III entirely, the piles become (1, 1, 0) -- two single-chip piles left, an even count with Anita to act -- so Anita empties one of them (say Pile I), leaving (0, 1, 0): one single-chip pile with Brinda to act, an odd count, so Brinda is fixed to take that last chip.

• If Brinda takes the single chip from Pile I (or symmetrically Pile II), the piles become (0, 1, 3) or (1, 0, 3); Anita responds by emptying Pile III entirely, leaving (0, 1, 0) or (1, 0, 0) -- again one single-chip pile with Brinda to act, so Brinda is fixed to take the last chip.

• If Brinda takes one chip from Pile III, the piles become (1, 1, 2); Anita takes one chip from Pile III as well, leaving (1, 1, 1) -- three single-chip piles, an odd count with Brinda to act, so Brinda is fixed to be the one who eventually removes the final chip.

In every one of Brinda's replies, Anita can force the count of remaining single-chip piles to be odd on Brinda's turn, so Brinda always ends up taking the last chip. Anita should therefore draw exactly one chip from Pile II.

Cross-check

Checking Anita's other possible opening draws (each also taking exactly one chip, matching the question's constraint) confirms Pile II is the only winning choice. Taking the one chip from Pile I leaves (0, 2, 3); Brinda can then empty Pile II entirely, leaving (0, 0, 3) with Anita to act, and playing that single pile out chip by chip leaves Anita taking the final chip. Taking the one chip from Pile III leaves (1, 2, 2); Brinda can empty Pile I, leaving (0, 2, 2) with Anita to act, and that position again forces Anita to eventually take the last chip. Only the opening draw of one chip from Pile II keeps the odd single-chip-pile parity on Brinda's side.