© Copyright 2005 by James D. Allen

Twelve Ancestors -- Solution

From general knowledge we know that every person has a male father and a female mother, and is not his or her own ancestor. The twelve include five women and their husbands (1) and two other men (2). None of the twelve has any children who are half-siblings (2); if someone's mother is in the set of 12 then so is that person's father (1). [The clue that Aella, etc. are not half-brothers (3) is not necessarily redundant because their parents need not be included in the set of twelve. Clue (3) doesn't clarify whether the four men who have brothers in the set can be a foursome of all brothers, but we will prove that that case is impossible.]

It is convenient to denote people by their ancestor numbers relative to Nathan; thus Nathan's mother is #3 and her mother is #7. [In the ancestor numbering system, the father of #N is #(2N) and his/her mother is #(2N+1).] Ancestors #3, #5 and #7 are in the set (6) and distinct [#3 cannot be #7 (mother); #5 cannot be #3 [she'd be married to her own son] or #7 [their daughter would be married to her own brother] (2).

At most one of the grandparents is descended from another grandparent [otherwise there would be a forbidden self-ancestor or sibling marriage], so at least one grandparent pair will have mothers distinct from #3, #5 and #7 and distinct from each other [else sibling marriage]; this accounts for all five women, and, since their husbands are included in the set (1), all of the 12 ancestors except Ephraim and Fjolnar (2). That Ephraim and Fjolnar are ancestors is given explicitly (4,5).

None of the grandparents is descended from another grandparent [all four of his or her grandparents would be in the set (above); these would be Nathan's great great grandparents but the set contains only three of them (6)]. One grandparent pair has four parents in the set (above); the other grandparent pair must have the same four [all husband-wife pairs are accounted for, but all great grandparents are in the set (6) and none of Nathan's grandparents is also his great-grandparent (above)]; thus Nathan's parents were ``double cousins.''

The parents of the four great-grandparents must include exactly three people in the set (6); since only two people [Ephraim and Fjolnar] are as yet unaccounted, these great great grandparents must include at least one of the great-grandparents; each of these had children only by one spouse (2), so that spouse is also one of the three great great grandparents. This means that couple had two grandchildren who married their own uncles [not aunts (2)]; in the ancestor table therefore we have #12 = #8, #13 = #9, #14 = #10, #15 = #11; and furthermore #8 and #9 are the parents of #10 or #11.

#11 is ancestral to three of the other women [#3, #5, #7] so cannot be descended from Fjolnar (4).

Ancestors #8 and #9 are not the parents of #10 [else #4, #6, #10 would form a brother trio, so there would be a brother quartet (3); the fourth brother could not be #2 or #8 so would be Fjolnar or Ephraim and would have to be ancestral to #11 [else could not be placed as an ancestor], so not Fjolnar (above) or Ephraim [#8 and #9 or their descendants couldn't be his grandchildren if he is their son, #5 and #7 couldn't be his grandchildren [#11 would be his daughter which is forbidden (5)] so his three grandchildren couldn't be placed]] so they are the parents of #11; we can add #8 = #22, #9 = #23 to the ancestor table.

From the ancestor table we see that Ephraim and Fjolnar must each be ancestral to #8, #9 and/or #10. #8 and #9 are #11's parents so Fjolnar isn't their ancestor (above); he is ancestral to #10 and must be #10's father [he has a granddaughter in the set (4)]. Fjolnar is the third of Nathan's great great grandparents (6). He isn't brother of #4 [#11 would have married her nephew which is forbidden (2)], so there is no brother quartet; #4 and #6 are one brother pair and the other pair is two of #8, #10, #20 and Ephraim [only possibilities].

Ephraim isn't the father of #8 or #9 [he would be a fourth great great grandparent which is forbidden (6)] so isn't the grandparent of #2, #3, #4, #6, #11 or himself and isn't the grandparent of #5 or #7 [only slot unassigned is #10's mother but Ephraim is male] so his three grandchildren are drawn from the set #8, #9, #10, #20, but cannot include both #10 and #20 [neither #21 nor #41 is Ephraim's daughter (5) which would be the only way] so include #8 and #9; Ephraim had no daughters (5) so has ancestor numbers #32 and #36.

Either #10 or #20 is also Ephraim's grandchild [he has three altogether (5) and all other possibilities are accounted], so Ephraim's brother, if any, isn't #10 [would be self-ancestor] or #20 [#10 would be Ephraim's grandson [elimination] but Ephraim had no daughters (5) so Ephraim's brother would be his own son] or #8 [grandson]. Ancestors #8 and #10 are not brothers [#20 [Fjolnar] would be ancestral to #8 so #11 which was ruled out (above). #20 is not his son's brother, so #20 and #8 are brothers [only possibility left]; so #40 = #16, #41 = #17. Fjolnar is the third of Ephraim's grandchildren.

Fjolnar is the unnamed brother so #8 is Gregor (3). Aella and Donogh are #4 and #6 in some order (3), so Benno is #2 and Donogh is #6 (7); Aella is #4. Cadelon is #10 [elimination]. Ancestor #9 has only one daughter [#11] in the set, who is Sabina or Wilmot (4) so not Valpurge, so #9 isn't Wilmot and must be Sabina (4). Wilmot is #11; Tamar is #3 and Valpurge is #7 (7). Ursula is #5 [elimination].

Note, by the way, that Ephrain is the agnatic ancestor of each of the 12 people; that is, if male, he inherits his surname and Y-chromosome from Ephraim; if female, she inherits her maiden surname from Ephraim and her father has Ephraim's Y-chromosome.

Here is the solution summarized as an ancestor table:

 1 Nathan (Alphonso XII King of SPAIN) 2 Benno (Francisco de Asis BOURBON) 3 Tamar (Isabel II Queen of SPAIN & the Indies) 4 Aella (Francisco de Paula BOURBON) 5 Ursule (Louise BOURBON of the Two Sicilies) 6 Donogh (Ferdinand VII King of SPAIN & the Indies) 7 Valpurge (Cristina BOURBON of the Two Sicilies) 8 Gregor (Charles IV King of SPAIN) 9 Sabina (Marie-Louise of PARMA) 10 Cadelon (Francis I BOURBON of the Two Sicilies) 11 Wilmot (Isabel BOURBON) 12 same as 8 13 same as 9 14 same as 10 15 same as 11 16 NN (Charles III King of SPAIN) 17 NN (Marie-Amelie of SAXONY) 18 NN (Felipe BOURBON Duke of Parma) 20 Fjolmar (Ferdinando I King of SICILY) 22 same as 8 23 same as 9 32 Ephraim (Philip V King of SPAIN) 33 NN (Isabella Elizabeth FARNESE) 36 same as 32 37 same as 33 40 same as 16 41 same as 17

(It turns out that this is the actual official pedigree of an 18th century Bourbon King. Above I show the names from that pedigree in parentheses.)

(Here is the Puzzle.)