# Background

I am researching genealogy on 23andMe.com, trying to find new cousins and determine their relationships to me. All the site gives me is a list of people and their predicted cousin-relationship to me (downloadable as a CSV file), along with a page where I can view a list of common relatives between us and a comparison of their predicted relationship to them (not downloadable, e.g. on person X's page it will say, "person Y-- You: 2nd cousin, X: 3rd cousin"). Unfortunately, in my culture, there has historically been a significant amount of endogamy due to discrimination, so "Distant cousin" can probably mean 10th+ cousins, or more. For my purposes, I take it to mean unrelated. This is the best I can do, given that 23andMe does not give access to a structured data set of all common relatives with respect to a given person, thereby removing the possibility of ruling out relatedness based on the lack of a relative being in another relative's common relatives set. Hence, if I see on person Z's profile that they are distant cousins to several of my cousins from my father's side, and they are 3rd cousins to person A who I'm unsure about, I will reason that person Z is on my mother's side, as is person A.

# Problem

Reasoning about the information in the form of "If you are 2nd cousins with X and distant with Y, Z, A, and B, and X is 3rd cousins with C, 4th with D, and distant with E, then maybe the connection is..." has become exceedingly difficult and has not produced a smidgeon of results.

# Question

Is it possible to determine the exact person-to-person relationship to a cousin solely based on your relationship to mutual cousins vs. their relationship to mutual cousins, and those cousins to other cousins, etc.? If so, how would one go about determining the relationship?

If that was not clear, here is an example to illustrate the question:

Let's say that me and 3rd cousin X do not know how we are related, but share cousin A: X is 2nd cousins with A, I am 4th cousins with A. X is also 3rd cousins with shared cousin B, and B is 2nd cousins with (shared) cousin C who I know is my 2nd cousin from my father's side, etc. By going through this process enough times and documenting the information as I go or graphing it out on a tree, can I deduce that, for example, cousin X must be my paternal grandmother's paternal cousin? Is that possible? How can I do that?

# Progress so far

Sometimes you learn more from what you don't know than from what you do, and often it makes the future research direction clearer when you get clear on what you don't know, so here is

### What I don't know

After long consideration of the problem, the conclusion I came to was that the following four questions are at the heart of whether the question can be answered and how it would be done (maybe I am wrong and these are not at the heart of it):

1. How do I determine whether two people come from the same grandparents?
2. How do I determine from which side each person descends from those grandparents? (i.e. X's maternal grandfather and Y's maternal grandmother)?
3. How do I keep track of those relationships (especially important if I were to write a program to do it)?
4. In general, how much can you really determine from just knowing the cousin relationship to people? Are any of these things possible to determine?

### What I do know

What I have determined so far:

Given: A is nth cousin of C and B is nth cousin of C

• To confidently say that A, B, and C shares the same nth-great-grandparents: A and B must also be nth cousins with each other.

But rarely does that happen, and thus, this rule does not take me very far. It is also not yet helpful because it does not explain how to determine which side the common grandparents are from.

# The monotony of the solution is not a barrier

Even if the answer to my question is a method which involves a lot of trial and error, monotony, etc, so long as it is rigorously methodical (in a mathematical sort of sense,) such that will work 100% of the time when followed through to the end, I can teach it to a computer and send those results to my hand-made graph generating program by calling a few lines of code like this:

``````    graph.addEdge("Grandmother #1" ,"Father #1")
``````

This is what the above lines create:

Sometimes "working 100%" will mean determining that something cannot be determined. That's ok. I'd like to assume though that with enough cross-referencing, the whole picture can be figured out. I'd assume that this program would be useful to many people, so if I get help creating it I would love to distribute it so that others can benefit from it.
P.S: I wrote some of

• One problem is that a prediction from 23andMe (or any other DNA site) of 3rd cousins doesn't rule out them actually being 2C2R -- same DNA range, utterly different diagram. DNA simply can't tell some relationships apart. Not even the "100%" predictions are necessarily true, because they fail to take identical twins into account. Aug 19 '20 at 19:04
• Aha. I now understand the premise of "What Are The Odd" - the information is probably not correct as it is being delivered. Most often, 23andMe doesn't suggest "X removed", except for very close cousins which you have confirmed the relationship on their tiny "Family Tree Builder"..
– Sam
Aug 19 '20 at 19:08

It sounds like DNA Painter's What Are The Odds (WATO) tool might be of help. It helps you figure out how a target person is related to a set of other people who's relationship is already known, based on their DNA. You can read more about it here.

There is also a version 2 that's currently in beta, but I can't speak to the new features it has.

• Although an interesting tool, it requires write a bit of information that I don't have. Most of the relatives I have talked with so far don't have family trees connecting each other, and the requirement to know the common ancestors is precisely what I am seeking in this question! I will definitely use it for other things though. Thanks for the tip!
– Sam
Aug 18 '20 at 12:03
• The question explicitly states, "...solely based on your relationship to mutual cousins vs. their relationship to mutual cousins..." not the knowledge of a common ancestor or anything else.
– Sam
Aug 18 '20 at 12:22
• You don't need names to use WATO, but in this case it will take some guesswork. If you have a pair of 2nd cousins, you can make a WATO tree that includes anonymous great-grandparents, then use the hypotheses to calculate how likely/unlikely it is that some 3rd sample belongs in various places in that tree. I've done this in an effort to determine how my known ancestors are related to other people's known ancestors and have mostly gotten results that are negative (e.g., "not siblings"), but still useful. You don't need a known common ancestor to built a WATO tree. Aug 18 '20 at 18:00

Although WATO trees have been suggested (including by me) as your best hope, I don't believe that even they will be adequate to the task you propose. I'll go so far as to claim that it can't be reliably done using only DNA match data. Once you get beyond the close relationships of parent/child, sibling, and maybe 1st cousin, there are too many possible relationships that share the same cM range to reliably distinguish between them.

Suppose you were to build a tree like the one in your example, then do a pairwise comparison of each sample pair (N!/2 comparisons) and look up the probabilities using the Shared cM tool (and, btw, if you can implement something that does that automatically, you could probably sell it). As the number of samples goes up, the probability of error in the overall tree goes up faster, and while it may be possible to determine candidates for incorrect placement in the tree, you can never be certain that your highest-probability hypothesis accurately reflects any removes or half cousins (or double cousins - if, as you say, there's a lot of endogamy in your culture, then all matches are suspect).

This is why DNA genealogy has never been recommended as a replacement for traditional "paper trail" research, because it can't be, at least not in the foreseeable future. Even with chromosome browsers and tracing of segments across samples, all you can really prove is who has common ancestors, and possibly how far back.

I realize that my justification may suffer from what my colleagues and I call "proof by lack of imagination" (as in "I can't imagine how this could possibly work"), so if anyone can point out a significant flaw in my argument, please do so (it helps me as much as anyone else).