Nim Calculator Form
Use up to nine heaps. Zero means an unused pile.
Sample Plotly Graph
Before submission, the chart shows the classic sample position 3, 4, 5 and one optimal next state.
Example Data Table
These sample positions help verify the solver.
| Piles | Mode | Nim-sum | Expected Result | Example Best Move |
|---|---|---|---|---|
| 3, 4, 5 | Normal | 2 | Winning | Reduce Pile 1 from 3 to 1 |
| 7, 7 | Normal | 0 | Losing | No forcing move exists |
| 1, 1, 1 | Misère | 1 | Losing | No safe move exists |
| 1, 1, 4 | Misère | 4 | Winning | Reduce 4 to 1 |
| 2, 2, 5 | Normal | 5 | Winning | Reduce 5 to 0 |
Formula Used
Nim strategy relies on binary XOR. The calculator evaluates whether the current player can force a win.
nim_sum = p1 XOR p2 XOR p3 XOR ... XOR pn
For normal play, a position is losing when nim_sum = 0.
Winning move target: target_heap = selected_heap XOR nim_sum, when target_heap < selected_heap.
For misère play, use normal XOR strategy unless every active heap equals 1.
When all active heaps are 1 in misère play, odd heap counts are losing and even heap counts are winning.
This page also lists all forcing moves. Each winning move leaves the opponent in a mathematically unfavorable position.
How to Use This Calculator
- Choose the rule set: normal play or misère play.
- Select how many pile inputs you want visible.
- Enter each pile size. Use zero for empty or unused piles.
- Press Calculate Nim Position.
- Read the result section shown below the header and above the form.
- Review the Nim-sum, binary heap table, and all winning moves.
- Use the graph to compare the current position with the recommended next state.
- Download the result as CSV or PDF when needed.
FAQs
1. What is the Nim-sum?
The Nim-sum is the bitwise XOR of all pile sizes. It is the main value used to decide whether a position is mathematically winning or losing.
2. Why does a zero Nim-sum matter?
In normal Nim, a zero Nim-sum means the player to move is in a losing position if the opponent responds perfectly. Nonzero values usually allow a forcing move.
3. How is misère Nim different?
Misère Nim changes the ending rule. Taking the last object loses. Most positions still follow normal XOR strategy, except when every active heap has size one.
4. Can I enter empty piles?
Yes. Enter zero for unused heaps. The calculator automatically ignores them in the strategic analysis while still keeping your visible layout intact.
5. Why are several winning moves sometimes listed?
A Nim position can have more than one move that leaves the opponent in a losing state. The calculator lists every forcing option it finds.
6. Does pile order change the answer?
No. Only the pile sizes matter. Rearranging the same heap values does not change the Nim-sum or the underlying game state.
7. Can this handle large heap values?
Yes. Nim analysis scales well because XOR is efficient. Extremely large values still work, provided your server can process ordinary form submissions comfortably.
8. What does the recommended next position mean?
It shows the board after the best calculated move. That target state is designed to leave the opponent with a weaker mathematical position.