Question: Who is Ariq Böke's paternal grandmother?
Answer: Börte

Tolui, (Classic Mongolian: Toluy, Tului, , , Tolui Khan (meaning the Khan Tolui)) (c. 1191–1232) was the fourth son of Genghis Khan by his chief khatun Börte. His "ulus", or territorial inheritance, at his father's death in 1227 was the homelands in Mongolia, and it was he who served as civil administrator until 1229, the time it took to confirm Ögedei as second Great Khan of the Mongol Empire (1206–1368)