In genealogy, a pedigree chart, which shows one's direct ancestors (parents, grandparents, etc. but not siblings, cousins, etc.) is often replaced by the equivalent but space-saving Ahnentafel table.
An Ahnentafel table is simply a numbered list of each ancestor, usually on separate lines. The "root" person goes on line 1. Then, for any person on line n, his father goes on line 2n and his mother goes on line 2n+1. Every ancestor gets a unique line, and every line gets a unique ancestor* (mathematically, at least -- in real life Ahnentafels, because a person may not know all of his ancestors some lines may be blank, and in the case where cousins married, their common ancestors may show up in several places in their children's Ahnentafels).
Question 1: Your great-great-grandfather(2nd-great-grandfather) was the first of his name (surname) (which you inherited) to come to America. What is his Ahnentafel number? What is the Ahnentafel number of your nth-great-grandfather of the same name?(Assume the the Western tradition where a child inherits his father's surname)
Question 2: Your Mitochondrial DNA is passed on only from your mother, who got it from her mother,etc. What is the Ahnentafel number of the great-grandmother from whom it "originally" came? Of the nth-great-grandmother?
[Hint: for the general case (nth-great-grandfather in question 1, nth-great-grandmother in question 2) it might be easier to work with m=n+2; m is the number of generations between the ancestor and your children. For n=1 (your great-grandfather), m=3 -- three generations in between: your grandfather, your father, and you.]
*This statement (that there is a one-to-one correspondence between Ahnentafel numbers and the set of all natural numbers) is fairly easy to prove. And, in fact, the proof is part of a later puzzle in this series. For this puzzle, it can simply be assumed.
[I did not copy/paste this answer from the Internet. I reasoned it out after examining a 7 generation sample that I worked out.]
I use a handy notation of D's and M's, where for instance DMMDM is my Dad's Mom's Mom's Dad's Mom. If we consider relative positions of the D's in DMMDM, counting from the right, beginning with relative position 0, there are D's in relative positions 1 and 4. Then the Ahnentafel number for DMMDM is gotten by subtracting the 1st and 4rth powers of 2 from (2^5+2^4+2^3+2^2+2^1+2^0), which results in
Then the general rule for an n-deep path is to sum the powers of 2 from 2^n to 2^0, omitting those powers which correspond to the relative positions of the D's, starting at the right with relative position 0.
So the nth great grandfather is:
DDDDDD....D (n+2 D's)
and the formula for his Ahnentafel number is
The nth great grandmother is:
MMMMM....M (n+2 M's)
and the formula for her Ahnentafel number is:
The second great grandfather has Ahnentafel number 2^4 =16
The second great grandmother has Ahnentafel number = 2^4+2^3+2^2+2^1+2^0 = 31
(Which is one more than that of her husband, MMMD, who has Ahnentafel number = 2^4+2^3+2^2+2^1 = 30)
Now getting back to those Blute Brothers....
Edited on April 11, 2004, 8:40 pm
Posted by Penny
on 2004-04-11 16:52:48