Nim 21

Master the strategy to always reach exactly 21

Achievements & Badges

The 21 Game

Current Total

Player 1

Your Turn!

AI - Hard

Waiting...

Choose a number to add:

AI is thinking

Game Settings

Target Number: The number to reach (10-100)

Max Pick Per Turn: Maximum numbers to add per turn (1-10)

Starting Number: Starting total (0-50)

Win Condition: Choose how to win

Opponent: Choose your opponent

First Player: Who goes first

Theme: Visual theme

Enable Sound Effects Show Strategy Hints

Daily Challenge: Clear today's daily challenge completion to replay

Winning Strategy & Mathematics

The Key Formula

The winning strategy is based on modular arithmetic. The magic number is:

Magic Number = Max Pick + 1

For the default game (max pick = 3), the magic number is 4.

Winning Positions

You want to leave your opponent at positions that are multiples of the magic number:

How to Play Optimally

Calculate the remainder: Divide the current total by the magic number
If remainder is 0: You're in a losing position (if opponent plays perfectly)
If remainder is not 0: Pick that remainder to reach the next multiple
Keep forcing your opponent to multiples of the magic number

Example (Target 21, Max Pick 3)

Start at 0 - You're at a multiple of 4 (bad position)
If opponent picks 2 - Total is 2
You should pick 2 - Total becomes 4 (multiple of 4)
If opponent picks 3 - Total is 7
You should pick 1 - Total becomes 8 (multiple of 4)
Continue this pattern - Force opponent to 12, 16, 20, then 21 (you win!)

Current Game Winning Positions

Hint for Current Position:

Game Theory Connections

This game is a variant of Nim, one of the most important games in combinatorial game theory. It demonstrates:

Impartial games: Both players have the same moves available
Perfect information: No hidden information or randomness
Solved games: There exists a winning strategy for one player
Backward induction: Working backwards from the end to find optimal moves

Learn more about Nim on Wikipedia

Statistics

Games Played

Games Won

Games Lost

Win Rate

Move History

No moves yet. Start playing!

Game Over!

Final Total: 21

Online Multiplayer

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