Juusan Sai no Tanjoubi, Kougou ni Narimashita.

On her thirteenth birthday, Rian is sent to join the harem of the emperor of Sekisou Kingdom. However, the person receiving her is Akatsuki, who had usurped the throne through "legitimate means" to become the new emperor! Rian is made empress, as she is deemed "suitable," but even though they share a bed, Akatsuki treats her like a child, like a hot water bottle to warm his bed. To help Rian sleep, Akatsuki challenges her with a problem to solve every night!? (Source: Kadokawa, translated)