The riddle on the swing (Ghatotkacha's bride and Barbarika's birth)
Krishna proposes Kamakantakata, Mura's daughter, as Ghatotkacha's bride; he wins her in Pragjyotisha with a riddle and in battle, she carries him to Indraprastha, and their son is born grown, named Barbarika for his hair and Suhridaya by Krishna. Khatu gives the boy a mother called Morvi or the Naga princess Ahilavati, who teaches him war, and has Shiva give the three arrows and Agni the bow.
West India · Indraprastha, Pragjyotisha, DvarakaAll traditions
At Indraprastha Krishna proposes Kamakantakata, daughter of the demon Mura, as Ghatotkacha's wife. She once fought Krishna, until Kamakhya protected her; she marries only the man who beats her in debate and in battle, and many have died trying. Yudhishthira objects, Bhima and Arjuna urge, and Krishna sends Ghatotkacha, telling him to think of him in the debate. In a golden mansion outside Pragjyotisha he finds her on a swing and asks a riddle she cannot answer. Her rakshasas and beasts from the swing chain are swallowed by twice as many of his; he throws her down. Beaten three ways, she carries him on her back to Indraprastha, where they marry by Kuru and rakshasa rites. Their son is born grown, cloud-dark, with bristling hair; Ghatotkacha names him Barbarika. At Dvaraka Krishna names him Suhridaya and sends him to the goddesses at Guptakshetra.
Khatu calls him Morvinandan, son of Morvi; the temple katha names his mother Ahilavati, a Naga princess. She teaches him war. He does tapas to Shiva, who gives him three arrows that cannot fail, one to mark targets, one to mark lives to spare, one to destroy the marked; Agni gives him a bow that can win the three worlds. Some tellings give the arrows to Durga or the goddess Vijaya instead, and many say Krishna sent him to the hidden shrine at the Mahi sangam to worship the nine Durgas.
What is different here:
No riddle or courtship; a mother who teaches him; Shiva's arrows and Agni's bow.
Sources: Skanda Purana, Maheshvara Khanda, Kaumarika Khanda 59, The departure of Ghatotkacha to Pragjyotisha, tr. G. V. Tagare (Wisdom Library); Skanda Purana, Kaumarika Khanda 60, The birth of Barbarika, tr. G. V. Tagare (Wisdom Library); Skanda Purana, Kaumarika Khanda 61, The liturgy of Ganeshvara, tr. G. V. Tagare (Wisdom Library); Barbarika Upakhyana, Skanda Purana 59 to 66, chapter summary (dvaipayana.net); Barbarika (Bharatkosh, bharatdiscovery.org, citing Bharatiya Sanskriti Kosh vol. 2, ed. Devendra Mishra, p. 525); Khatu Shyam katha (khatu.in); Khatu Shyam katha (bhaktibharat.com); Barbarika (Wikipedia)