Simple code to compute the values of the Hydra game from Numberphile.

HydraGame.cs

// Reference here: https://www.youtube.com/watch?v=prURA1i8Qj4

var rightMost = true;

var step = BigInteger.Zero;

var initialDepth = 5;

// Represents a simple tree (single trunk). Length is tree depth. Root node is excluded. Value is count of leaf nodes (heads) at that depth.
// Nodes will always be added to the right of the implied connections.
var nodes = Enumerable.Range(0, initialDepth).Select((_, ix) => ix == initialDepth - 1 ? BigInteger.One : BigInteger.Zero).ToList();

var timeLimit = TimeSpan.FromMinutes(0.3);
var sw = Stopwatch.StartNew();

while (nodes.Count > 0 && sw.Elapsed < timeLimit)
{
    step++;

    int ix;
    if (rightMost)
    {
        // Remove heads from the level that has most recently had heads added.
        // Assuming that, as above, nodes are added left-to right, this corresponds to the lowest-index non-zero cell.
        for (ix = 0; ix < nodes.Count - 1; ix++)
        {
            if (nodes[ix] > 0)
            {
                break;
            }
        }
    }
    else
    {
        // Remove the furthest-away head.
        ix = nodes.Count - 1;
    }
    
    var remaining = nodes[ix] = nodes[ix] - 1;

    // Add new heads to the grand-parent node, n = step.
    if (ix > 0)
    {
        // OPTIMIZATION: Adding to the root node just adds more steps, effectively.
        if (rightMost && ix == 1)
        {
            // OPTIMIZATION: Don't add nodes. Instead, Add steps based on the nodes we would add, i.e. double the step number.
            step *= 2;
        }
        else
        {
            nodes[ix - 1] += step;
        }
    }

    // When all heads are removed from a neck, it becomes a single head (leaf node).
    if (remaining <= 0 && ix == nodes.Count - 1)
    {
        nodes.RemoveAt(ix);
        if (nodes.Count > 0)
        {
            nodes[^1] += 1;
        }
    }
}

step.ToString().Dump();
if (nodes.Count > 0)
{
    nodes.Dump("INCOMPLETE RESULTS");
}