What is the basis upon which the Mushaf was divided into parts and hizbs, and why does the size of the quarters of the hizbs vary?

The division of the Mushaf into ajza' (parts), ahzab (sections), and arbāʿ (quarters) is a matter of ijtihad (independent reasoning). The well-known tahzīb (partitioning) among the Companions was three surahs, then five, then seven, then nine, then eleven, then thirteen, then the Ḥizb al-Mufaṣṣal alone. The Uthmanic Masahif (copies of the Quran) were devoid of any such divisions.

Dividing the Mushaf into thirty ajza', or dividing each juz' into two ahzab, and each hizb into four arbāʿ, or placing the words "khams" (five) and "ʿashar" (ten) at the ends of verses, are all later ijtihādāt.

The tahzīb found in today's Masahif is based on the number of letters. It is reported that Al-Hajjaj ibn Yusuf al-Thaqafi ordered this. This letter-based tahzīb was an innovation from the era of Al-Hajjaj in Iraq, whereas the people of Madinah did not know of it. The Companions estimated their recitation by verses or surahs.

Ibn Taymiyyah held the view that tahzīb by complete surahs is the best, because it does not involve stopping at words connected to what follows them, and because it was the prevailing custom of the Prophet (peace be upon him) and his Companions. He also noted that the newly introduced division does not allow for equality in the number of letters across the ajza'.

The Permanent Committee issued a fatwa stating that there is no evidence for the tahzīb currently marked on the margins of Masahif, and that what is authentically reported from the Companions is the surah-based tahzīb mentioned earlier. In essence, the tahzīb of Masahif today relies on the number of letters, which differs from the optimal tahzīb followed by the Companions based on surahs. The matter, however, is easy.