Seventh case: The grandfather’s three alternatives—equal division (mukāsama), one-third of the residue, or one-sixth of the estate—turn out to be identical. An example is: a husband, a grandfather, and two brothers, a combination in which the fixed shares come to one-half of the estate and the brothers are twice his number.

Why the three amounts coincide is this: once the husband’s half is taken, the other half remains for the grandfather and the two brothers. Within that half, one-third of the residue, the result of equal division, and one-sixth of the whole estate all equal the same fraction. Because one-third of the residue cannot be expressed exactly on the original denominator, we multiply the divisor of the third (3) by the original root of the problem (2), producing 6.

From that new denominator the husband receives his original share—one part—multiplied by 3, giving 3 shares and leaving 3. Of those 3 the grandfather receives 1 share in every case, and the remaining 2 go to the two brothers, one each1.

Schematic diagram:
Base 2 6
Husband 1 3
Grandfather — 1
Two brothers — 2 (1 each)

Note
With respect to what may remain over and above the grandfather’s fixed share—whether any residue exists or not—there are four situations: