The past couple of weeks, this image has been circulating on Facebook, entitled the Cousin Explainer.

Given the huge number of likes (256, at time of writing this) and shares (3,400, at time of writing this), it seems I wasn't the only one who needed this mapping out clearly.

As a mathematician, it always bothered me that I could not explain, or even better 'define', what an mth cousin n-times removed is. Well, now I can. And I share it here so that you can too.

## Is "First Cousin Once Removed" ambiguous?

As various friends of mine have shared this image, there is one question that seems to come up repeatedly:

"I wasn't aware that there were two ways to be first cousin once removed."

The answer is: There is, and there isn't, two ways to do this.

I'll come to a more general definition in a moment. But, for now, note that the individual on the diagram ("you"), has two first cousins once removed.

This is not because "first cousin once removed" is not well-defined. In each case, "you" and your first cousin once removed are related in the same way. One of you has a parent that is brother or sister of a grandparent of the other one.

In one case, it's your parent and your first-cousin-once-removed's grandparent who are siblings. In the other case, it's your grandparent and your first-cousin-once-removed's parent.

To put it another way: The relationship of "first cousin once removed" has to be reciprocal, or "commutative" (to use the mathematical term). If I'm your first cousin once removed, then you must also be my first cousin once removed.

That only creates ambiguity you don't get with simple "cousins" because your first cousin once removed will be one generation above or below you.

Now we've got that out of the way, let's define this precisely.

## Define "Mth Cousin N Times Removed"

Suppose two people, X and Y are Mth cousins N times removed. What does this mean?

We're only defining this for M and N ≥ 0.

### Assume, for now, that M and N are both strictly greater than 0.

This means that somewhere in the generations above X and Y are a pair of siblings, X_{0} and Y_{0}. So, X_{0} and Y_{0} are brother or sister. X is descended from X_{0}. Y is descended from Y_{0}.

Assume, without loss of generality, that X is either the same generation as Y, or is in an older generation.

For them to be Mth cousins N times removed, X is M generations below X_{0}, and Y is M+N generations below Y_{0}.

### Now we can deal with the cases where M = 0 and/or N = 0.

**If N = 0, and M > 0**: X and Y are simply what we know as Mth cousins. You don't say they're, for example, "2nd cousins no times removed", you just say "2nd cousins". The definition above tells us what this means: It means that if you each go up the family tree by M generations, you hit two people who are siblings. So, 2nd cousins have grandparents who are brother or sister, 3rd cousins have great-grandparents who are brother or sister, and so on.

**What about if M = 0?** What are "zeroeth cousins", or "zeroeth cousins N times removed"?

**Well, let's take the case of M = 0 and N = 0**, just common-or-garden zeroeth cousins. Apply the definition above: This means that if you go 0 generations up the family tree, and your zeroeth cousin goes (0 + 0 = 0) generations up the family tree, you hit a pair of siblings. Going 0 generations up the family tree just leaves you with the person you started with. So "zeroeth cousins" are what we normally call brothers and sisters!

**If M = 0 and N > 0**. Again, apply the definition, and remember that "zero generations up the family tree" leaves you with the person you started with. Either your "zeroeth cousin N times removed" is directly descended from your brother and sister, coming N generations below them. Or you are N generations directly below their brother or sister. In other words these are (maybe "great") nephews, nieces, uncles and aunts.

A "zeroeth cousin N times removed" (N > 0) is either your (great) ^{(N-1)} niece or nephew, or your (great) ^{(N-1)} uncle or aunt.

## Articulate a Relationship in Terms of Cousins

Lastly, let's turn this round the other way.

We now know that Mth cousins N times removed are people with a pair of siblings somewhere in their ancestry. Suppose I know that I have a relation, and we are both descended from a pair of siblings. How can I work out what kind of cousins we are?

Well, let's say you are X generations below this common sibling pair, and your relation is Y generations below.

**If X = Y = 0**, then your relationship is that of brother or sister.

**If X = 0, but Y > 0**, then your relation is your (great) ^{(Y-1)} niece or nephew.

**If X > 0 and Y = 0 **then your relation is your (great) ^{(X-1)} aunt or uncle.

Lastly, **if X > 0 and Y > 0**, then you are min(X, Y)th cousins, | X – Y | times removed.

## For Further Study

What if three people, X, Y and Z are related, such that X and Y are cousins (of some kind), and Y and Z are cousins (of some kind)? What is the relationship between X and Z?

Well, this gets impossibly complicated if you bring relations by marriage into the equation. That's to say, X and Y could be cousins because X is descended from Y's dad's sibling. But Y and Z could be cousins because Z is descended from Y's mum's sibling. In that case, X and Z would not be related at all, at least not because they're both cousins of Y.

So we'd have to assume that this scenario is not in play. Assuming that, what can we say about the relationship between X and Z. There's some fancy group theory that could be done.

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