~ where perfect information allows for flawless strategy ~

A pile of 21 chips on a table

Combinatorial games serve as fascinating playgrounds for strategy and mathematical insight. But what makes a game "combinatorial"?

  1. Perfect information: All players know all the details of the game at all times.
  2. No chance moves: The game's outcome is determined solely by the players' choices, without the involvement of luck or chance.
  3. Zero-sum outcome: The game has a clear win, lose or draw.

The term "combinatorial" suggests a focus on the combinations of moves and states that can occur within the game. Mathematicians, computer scientists, and game theorists analyze these games to understand their structure, create algorithms that find optimal moves, and even solve them—that is, mathematically determine the outcome given optimal play from both sides.

A beetle saying life's a game


One game that fits these parameters perfectly is the 21-Chip Takeaway Game.


  1. Start with a pile of 21 chips.
  2. Players take turns to remove 1, 2, or 3 chips from the pile.
  3. The player who picks up the last chip wins the game.

Is there an optimal strategy? Let us know yours in the comments below. How does it change given different variables? Can you come up with a universal strategy?