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Two of the photographs show Louis in a woodland setting with moss stuck to his red sweater, while in a third he sports a blue jumper with a picture of a dog on the front.
Louis has largely been kept out of the public eye since he was born.
All were taken this month by Kate at the family's home in Norfolk, eastern England, Kensington Palace said.
Louis, who is fifth in line to the throne, has a five-year-old brother, Prince George, and three-year-old sister, Princess Charlotte.