Play Nim Online

Charles L. Bouton published a complete mathematical theory of Nim in 1901. He proved that players can classify any normal-play position by taking the bitwise exclusive OR, or XOR, of its heap sizes. A zero nim-sum marks a position that loses against correct play; a nonzero nim-sum gives the next player a route to zero. This makes Nim a rare game with a compact, proven decision method rather than a rule of thumb. Play free against the computer, test the XOR strategy, and learn why one careful removal can control the result.

1. Choose three, four, or five heaps, select normal or misere play, and set the computer difficulty. Each heap then shows how many objects remain.
2. On your turn, choose exactly one nonempty heap. Remove one object, several objects, or the entire heap from that chosen heap.
3. Leave every other heap unchanged. A move that removes objects from two heaps at once breaks the central rule.
4. Pass the turn after your removal. The computer chooses one heap and removes at least one object on its turn.
5. Continue until no objects remain. The player who takes the final object wins in normal play and loses in the selectable misere mode.
6. For strategic play, write each heap size in binary and XOR the values column by column. Equal binary digits cancel, while an unmatched 1 remains.
7. When the nim-sum is nonzero, seek a legal reduction that makes the new XOR equal zero. Repeat that response whenever the opponent returns a nonzero position.

• A position contains two or more separate heaps of countable objects.
• Each turn removes at least one object from exactly one heap. A player may remove every object in that heap.
• Players cannot add objects, move objects between heaps, skip a turn, or remove from multiple heaps.
• Normal play awards the win to the player who removes the final object from the final nonempty heap.
• The nim-sum equals the bitwise XOR of all heap sizes. XOR produces 0 when a binary column contains an even number of 1 digits and 1 when it contains an odd number.
• Bouton's result states that a zero nim-sum loses for the player whose turn begins there when both players choose correctly. Every nonzero position has at least one move to a zero nim-sum.
• Some variants reverse the ending or limit removals. Those changes require different analysis, so apply Bouton's normal-play result only to standard unrestricted Nim.

• Calculate the nim-sum before moving. For heaps 3, 4, and 5, compute 3 XOR 4 XOR 5: binary 011 XOR 100 XOR 101 equals 010, or 2. Since the result is nonzero, a winning reduction exists.
• Move to zero, not toward equal-looking heaps. From 3, 4, 5, reduce the heap of 3 to 1. The new values 1 XOR 4 XOR 5 equal 0. Visual symmetry can help sometimes, but XOR gives the reliable target.
• Use binary place values rather than ordinary addition. XOR ignores carries. Align 1, 2, 4, 8, and higher columns, then mark whether each column contains an odd number of 1 digits.
• Recalculate after every opponent move. A move from a zero position must create a nonzero nim-sum, so one fresh calculation identifies a response that restores zero. Do not repeat a memorized removal without checking the new heaps.
• Handle the one-object ending directly. When every nonempty heap has size 1 under normal play, parity decides the result: an odd number of heaps wins for the player to move, while an even number loses. XOR expresses the same fact immediately.

• Misere Nim: The player who takes the final object loses. Standard XOR strategy applies while at least two heaps exceed one object, but the all-ones ending requires a parity adjustment.
• Subtraction Nim: Rules restrict each move to a specified number of objects, such as one, two, or three. Modular patterns often replace unrestricted Nim's direct heap reduction.
• Single-Pile Nim: Players share one heap and face a removal limit. This version teaches backward reasoning, though it lacks the multiheap XOR structure that Bouton analyzed. Try Tower of Hanoi for another puzzle with a proven optimal pattern.

• Turns stay simple: choose one heap and remove any positive number of objects.
• The XOR result gives each standard position a verifiable mathematical status.
• Beginners can play by intuition, then improve by learning binary calculations one column at a time.
• Short rounds reveal mistakes quickly because every removal changes a small, visible state.
• Difficulty scales naturally through more heaps and larger values without adding exceptions to the core rules.

Play Nim online for free against a computer in your browser. Choose a heap, remove one or more objects, and try to take the final object under normal-play rules. No board setup or download is required. Start with small heaps to see the zero nim-sum pattern, then raise the difficulty and practice binary XOR on larger positions. Every turn gives you a fresh mathematical decision.

Nim pairs one plain move rule with a complete mathematical strategy. Remove objects from one heap, aim to take the last object, and use XOR to separate winning positions from losing ones. Bouton's 1901 result gives you a concrete target: leave a zero nim-sum after each turn. Start a free browser game, calculate the heaps, and test the proof against the computer.

• Nim, A Game with a Complete Mathematical Theory - JSTOR
• Bouton's 1901 Nim Paper PDF - University of Alberta