The Torch, And What He Kept

In Apollodorus, the solution is not strength. Iolaus brings fire, and as each head comes off he sears the stump so nothing can grow back — cutting and cautery together, neither working alone. The immortal head is buried under a heavy rock. Then Heracles does the thing that outlives the labour: he dips his arrows in the Hydra's venom. Those arrows go on to poison the centaur Chiron, who cannot die and must be released from immortality to escape the pain, and eventually to kill Heracles himself through the shirt of Nessus. He won by taking the venom with him, and it finished him in the end.