What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.
Which hour coincides with the last third of the night?
This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.
What is the ruling on one who vowed not to speak to a certain person, then forgot his vow and spoke to him?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Whoever swears an oath or makes a vow to abstain from something has a choice: if he wishes, he may keep to his vow, or if he wishes, he may offer the expiation of an oath (kaffārat yamīn). The Prophet (peace and blessings be upon him) said: “The expiation for a vow is the expiation for an oath.” Narrated by Muslim. This type of vow is called a nadhr al-lajāj (a vow made in stubbornness or anger).
If a person makes a vow and then forgets it, nothing is due from him, for forgetfulness is among the excuses that lift legal responsibility (taklīf) from a Muslim. This is based on the saying of the Prophet (peace and blessings be upon him): “Actions are judged only by intentions, and every person will have only what he intended.” [Narrated by al-Bukhārī]. There is no intention for the one who forgets or errs. It is also based on the report: “Indeed, Allah has pardoned for my Ummah mistakes, forgetfulness, and what they are compelled to do.” Narrated by Ibn Mājah and others, and graded authentic by Ibn Ḥibbān and al-Ḥākim.
It is stated in ʿUmdat al-Sālik (p. 243), one of the books of the Shāfiʿī school: “Whoever swears, [for example], ‘I will not enter the house,’ and then enters it forgetfully, ignorantly, under compulsion, or being carried, he does not break his oath.” And Allah Almighty knows best.
How do I deal with whispers (waswasa) in ablution and purification?
Whispers in ablution are from Satan, and the Muslim should not pay attention to them. He is not required to re-perform ablution or repeat washing a limb because of waswasa. He should always base his purification on its being valid. It is recommended for him to frequently say "La ilaha illa Allah" (There is no god but Allah), because Satan recoils when Allah is mentioned. And Allah the Almighty knows best.