Sixth case: the grandfather’s fixed one-sixth of the estate equals one-third of the residue. An example is: a husband, a grandfather, and three brothers—a combination in which the prescribed shares come to one-half of the estate and the brothers outnumber him by more than double.

Why the sixth and one-third of the residue coincide is this: once the husband’s half is set aside, the other half remains for the grandfather and the three brothers. One-sixth of the whole therefore represents exactly one-third of that remainder. The remainder, however, does not admit of an exact third, so 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 these 3 the grandfather takes 1 share, which is one-third of the residue and thus one-sixth of the entire estate. The brothers receive the remaining 2 shares, but because those 2 shares are to be divided among three heads, they neither divide nor correspond. We therefore multiply the corrected denominator (6) by the number of brother-heads (3), reaching 18.

The final distribution is as follows:
• Husband: 3 × 3 = 9 shares
• Grandfather: 1 × 3 = 3 shares
• Brothers: 2 × 3 = 6 shares — 2 shares each

Schematic diagram:
Base 2 6 18
Husband 1 3 9
Grandfather — 1 3
Three brothers — 2 6 (2 each)