What is the Hydra game?
The Hydra game is a mathematical game played on a finite rooted tree. The tree is imagined as a Hydra: its root is the body, its branches are necks, and its terminal vertices are heads.
Hercules repeatedly cuts off one head. In some versions, the Hydra responds by growing new branches. This means that cutting off one head can make the tree larger instead of smaller.
The exact regrowth rule determines the mathematical game.
Hydra game simulator
Configure an initial rooted tree and choose a simulation rule. The simulator will show the successive cuts and tree statistics.
Simulation results
The Hydra has been completely defeated.
Selected rules
Variant: Depth-descending regrowth
Head selection: Deepest head first
Initial structure: Height 3, branching factor 2.
Initial Hydra tree
Each node is identified by a number. The root is node 1. Indentation and connector lines show the parent-child structure.
Turn-by-turn history
The table records which head was cut, its depth, how many new heads were created, and the resulting tree size.
| Turn | Head cut | Depth | Regrowth | Nodes remaining | Leaves remaining | Maximum depth |
|---|---|---|---|---|---|---|
| 0 | — | — | Initial tree | 15 | 8 | 3 |
| 1 | Node 8 | 3 | 2 new leaf/leaves at depth 2 | 16 | 9 | 3 |
| 2 | Node 9 | 3 | 2 new leaf/leaves at depth 2 | 17 | 11 | 3 |
| 3 | Node 10 | 3 | 2 new leaf/leaves at depth 2 | 18 | 12 | 3 |
| 4 | Node 11 | 3 | 2 new leaf/leaves at depth 2 | 19 | 14 | 3 |
| 5 | Node 12 | 3 | 2 new leaf/leaves at depth 2 | 20 | 15 | 3 |
| 6 | Node 13 | 3 | 2 new leaf/leaves at depth 2 | 21 | 17 | 3 |
| 7 | Node 14 | 3 | 2 new leaf/leaves at depth 2 | 22 | 18 | 3 |
| 8 | Node 15 | 3 | 2 new leaf/leaves at depth 2 | 23 | 20 | 2 |
| 9 | Node 4 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 10 | Node 5 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 11 | Node 6 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 12 | Node 7 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 13 | Node 16 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 14 | Node 17 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 15 | Node 18 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 16 | Node 19 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 17 | Node 20 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 18 | Node 21 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 19 | Node 22 | 2 | 1 new leaf/leaves at depth 1 | 23 | 20 | 2 |
| 20 | Node 23 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 21 | Node 24 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 22 | Node 25 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 23 | Node 26 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 24 | Node 27 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 25 | Node 28 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 26 | Node 29 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 27 | Node 30 | 2 | 1 new leaf/leaves at depth 1 | 23 | 21 | 2 |
| 28 | Node 31 | 2 | 1 new leaf/leaves at depth 1 | 23 | 22 | 1 |
| 29 | Node 2 | 1 | No regrowth | 22 | 21 | 1 |
| 30 | Node 3 | 1 | No regrowth | 21 | 20 | 1 |
| 31 | Node 32 | 1 | No regrowth | 20 | 19 | 1 |
| 32 | Node 33 | 1 | No regrowth | 19 | 18 | 1 |
| 33 | Node 34 | 1 | No regrowth | 18 | 17 | 1 |
| 34 | Node 35 | 1 | No regrowth | 17 | 16 | 1 |
| 35 | Node 36 | 1 | No regrowth | 16 | 15 | 1 |
| 36 | Node 37 | 1 | No regrowth | 15 | 14 | 1 |
| 37 | Node 38 | 1 | No regrowth | 14 | 13 | 1 |
| 38 | Node 39 | 1 | No regrowth | 13 | 12 | 1 |
| 39 | Node 40 | 1 | No regrowth | 12 | 11 | 1 |
| 40 | Node 41 | 1 | No regrowth | 11 | 10 | 1 |
| 41 | Node 42 | 1 | No regrowth | 10 | 9 | 1 |
| 42 | Node 43 | 1 | No regrowth | 9 | 8 | 1 |
| 43 | Node 44 | 1 | No regrowth | 8 | 7 | 1 |
| 44 | Node 45 | 1 | No regrowth | 7 | 6 | 1 |
| 45 | Node 46 | 1 | No regrowth | 6 | 5 | 1 |
| 46 | Node 47 | 1 | No regrowth | 5 | 4 | 1 |
| 47 | Node 48 | 1 | No regrowth | 4 | 3 | 1 |
| 48 | Node 49 | 1 | No regrowth | 3 | 2 | 1 |
| 49 | Node 50 | 1 | No regrowth | 2 | 1 | 1 |
| 50 | Node 51 | 1 | No regrowth | 1 | 1 | 0 |
| 51 | Node 1 | 0 | No regrowth | 0 | 0 | 0 |
Latest tree structure
Supported Hydra variants
1. Cut-only Hydra
In this simple version, Hercules removes one leaf at each turn. No new nodes are created. The tree therefore becomes smaller until it disappears.
This is a basic tree-pruning exercise, not the famous regrowing Hydra game.
2. Depth-descending regrowth Hydra
This custom educational variant uses the depth of the cut head to determine regrowth.
- Cut a leaf at depth d.
- If d = 1, no regrowth occurs.
- If d is at least 2, create d − 1 new leaves.
- Attach the new leaves to the grandparent of the removed leaf.
- The new leaves are one level closer to the root than the cut head.
The regrowth depth decreases as the game progresses through newly created heads.
What is the Kirby–Paris Hydra?
The Kirby–Paris Hydra is a famous mathematical game involving finite rooted trees. Its rules allow the Hydra to regrow after Hercules cuts off a head. The regrowth is defined using copies of parts of the tree, with the exact behavior depending on the head's position and the particular formal version of the game.
The remarkable result is that Hercules eventually wins: every finite initial Hydra is eventually defeated under the standard game rules. Some strategies can nevertheless require an extraordinarily large number of moves.
Why are Hydra games connected to large numbers?
Certain formal Hydra games have extremely long termination times. Even relatively small starting trees can produce huge numbers of moves under their standard regrowth rules.
Their termination proofs are connected to ordinal notation and well-founded mathematical structures. This makes Hydra games an important example in proof theory and the study of fast-growing functions.
The key lesson is that a process can be guaranteed to terminate mathematically while still requiring far more steps than practical computation can simulate.
Frequently asked questions
What is the Hydra game?
The Hydra game is a mathematical game played on a finite rooted tree. A player repeatedly cuts off a head, represented by a leaf. Depending on the chosen Hydra rules, the remaining tree may grow new branches after a cut.
What is a Hydra in mathematics?
A mathematical Hydra is a rooted tree. The root represents the body, branches represent necks, and terminal vertices represent heads. The game changes the tree according to a specified cutting and regrowth rule.
Does every standard Kirby–Paris Hydra eventually die?
Yes. The Kirby–Paris Hydra theorem establishes that Hercules eventually defeats the Hydra under the standard rules, regardless of the initial finite Hydra and the allowed head-cutting choices.
Does this page implement the standard Kirby–Paris rules?
No. The page offers a cut-only variant and a custom depth-descending regrowth variant. The latter is designed for an accessible simulation and is not the standard Kirby–Paris rule.
What is a leaf in a rooted tree?
A leaf is a vertex with no children. In the Hydra game, a leaf represents a head that can be cut.
What does the depth of a head mean?
The depth is the number of edges between the root and the leaf. A head directly attached to the root has depth one.
Why can a Hydra grow after a head is cut?
Some Hydra rules specify that removing a head triggers regrowth elsewhere in the tree. This can create more heads even though Hercules has successfully cut one.
Can the Hydra game have more heads after a move?
Yes. In regrowth variants, a single cut can produce multiple new heads. The total number of nodes or leaves may increase temporarily.
Does a large number of moves mean the Hydra is infinite?
No. A finite Hydra can require an extremely large number of moves under certain rules. A long game does not mean the tree or the number of moves is infinite.
Why does the simulator stop at a limit?
The simulator uses maximum turn and node limits to protect server resources. Reaching a limit means the simulation stopped, not that the Hydra is mathematically impossible to defeat.
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