Question: Who is Ariq Böke's paternal grandmother?

Knowledge 1: Ariq Böke
Ariq Böke (after 1219–1266), the components of his name also spelled Arigh, Arik and Bukha, Buka (Chinese: 阿里不哥), was the seventh and youngest son of Tolui and a grandson of Genghis Khan.After the death of his brother the Great Khan Möngke, Ariq Böke claimed the title of the Great Khan of the Mongol Empire and briefly took power while his brothers Kublai (more commonly known as Kublai Khan) and Hulagu (more commonly known as Hulagu Khan) were absent from the Mongolian homeland.When Kublai returned for an election in 1260, rival factions could not agree, and elected both claimants, Kublai and Ariq Böke, to the throne, resulting in the Toluid Civil War that fragmented the Mongol Empire.Ariq Böke was supported by the traditionalists of the Mongol Empire, while his brother Kublai was supported by the senior princes of North China and Manchuria. [SEP] Knowledge 2: Tolui
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).Before that he had served with distinction in the campaigns against the Jin dynasty, the Western Xia and the Khwarezmid Empire, where he was instrumental in the capture and massacre at Merv and Nishapur.He is a direct ancestor of most of the Ilkhanids.Tolui never used the title of Khagan himself; neither Genghis Khan nor his immediate three successors would ever use any reigning titles unlike the neighboring Chinese dynasties in the south.Tolui was awarded the title of Khagan by his son Möngke and was given a temple name (Chinese: 元睿宗; pinyin: Yuán Ruìzōng; Wade–Giles: Jui-Tsung) by his other son Kublai, when he established the Yuan dynasty a few decades later.
Knowledge 1: Ariq Böke
Ariq Böke (after 1219–1266), the components of his name also spelled Arigh, Arik and Bukha, Buka (Chinese: 阿里不哥), was the seventh and youngest son of Tolui and a grandson of Genghis Khan.After the death of his brother the Great Khan Möngke, Ariq Böke claimed the title of the Great Khan of the Mongol Empire and briefly took power while his brothers Kublai (more commonly known as Kublai Khan) and Hulagu (more commonly known as Hulagu Khan) were absent from the Mongolian homeland.When Kublai returned for an election in 1260, rival factions could not agree, and elected both claimants, Kublai and Ariq Böke, to the throne, resulting in the Toluid Civil War that fragmented the Mongol Empire.Ariq Böke was supported by the traditionalists of the Mongol Empire, while his brother Kublai was supported by the senior princes of North China and Manchuria. [SEP] Knowledge 2: Kaoru Hatoyama
After the elections of 2009, she became more widely known as the grandmother of Prime Minister Yukio Hatoyama and his politician brother Kunio Hatoyama.
Knowledge 1: Ariq Böke
Ariq Böke (after 1219–1266), the components of his name also spelled Arigh, Arik and Bukha, Buka (Chinese: 阿里不哥), was the seventh and youngest son of Tolui and a grandson of Genghis Khan.After the death of his brother the Great Khan Möngke, Ariq Böke claimed the title of the Great Khan of the Mongol Empire and briefly took power while his brothers Kublai (more commonly known as Kublai Khan) and Hulagu (more commonly known as Hulagu Khan) were absent from the Mongolian homeland.When Kublai returned for an election in 1260, rival factions could not agree, and elected both claimants, Kublai and Ariq Böke, to the throne, resulting in the Toluid Civil War that fragmented the Mongol Empire.Ariq Böke was supported by the traditionalists of the Mongol Empire, while his brother Kublai was supported by the senior princes of North China and Manchuria. [SEP] Knowledge 2: Hannah Arnold
Hannah Arnold, also known as Hannah Waterman King, was the grandmother of Benedict Arnolds children.

Answer: Börte