First situation: after the fixed shares are distributed, more than one-sixth of the estate is left over; in that case the grandfather may choose whichever of the three options he prefers—equal division, one-third of the residue, or one-sixth of the whole estate.

Second situation: exactly one-sixth remains; that one-sixth goes to the grandfather as a fixed share.

Third situation: less than one-sixth remains; the grandfather’s share is increased until it reaches a full one-sixth.

Fourth situation: nothing at all remains because the fixed shares consume the entire estate; even so, the grandfather is still allotted one-sixth by way of augmentation.

In all three of these latter situations the brothers are excluded, except for the sister in the Akdariyyah case, as will be explained later.¹

Note
The grandfather is given one-third of the residue in certain cases by analogy with the mother in the two ʿUmariyyah situations, for each of them has a direct parental tie. Had no fixed-share heir been present, he would have received one-third of the estate; therefore, when the fixed-share heir takes his portion, the grandfather receives one-third of what is left and the remainder goes to the siblings. He is not granted the full third because that would harm the siblings. As for giving him one-sixth, the rationale is that his share never drops below one-sixth when descendants are present—a stronger claim—so it is even more appropriate when they are absent.²