In Solomon's judgment between the two women, to whom did he give the living baby?

Answer: The woman who pleaded for his life

The true mother begged Solomon to give the baby away rather than kill him. Solomon ruled, "Give the living baby to the first woman... she is his mother."

“Then the woman whose son was alive spoke to the king because she yearned with compassion for her son. “Please, my lord,” she said, “give her the living baby. Do not kill him!” But the other woman said, “He will be neither mine nor yours. Cut him in two!””— 1 Kings 3:26 (BSB)