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## It's All Relative

Published on Sunday, October 13, 2013 in , ,

Let's take a break from the usual math and memory fare to consider a different type of puzzle.

Over at the Futility Closet site, there's a great puzzle concerning relationships, titled “All Relative”:

A problem from Dick Hess’ All-Star Mathlete Puzzles (2009):

A man points to a woman and says, “That woman’s mother-in-law and my mother-in-law are mother and daughter (in some order).” Name three ways in which the two can be related.

This may seem confusing when you first read it, but it's not hard if you work through it bit by bit.

The first thing which should be made clear is that the woman's mother-in-law can be the mother or the daughter of the man's mother-in-law, so we'll need to work through both possibilities. Two possibilities also suggests only 2 answers, yet the puzzle suggests there are 3 answers, so we'll have to keep an eye out for other possibilities.

Since this puzzle involves in-laws, and therefore marriages, we're going to be assuming all marriages involved in this puzzle are heterosexual, as we'll be using Wolfram|Alpha to work through the family trees, and this assumption will make it easier for that site to process. As you can see by the examples here, Wolfram|Alpha can handle genealogy problems quite well.

Being a man myself, I'm going to work from the man's side of the family tree. The same logic and approach will work from the woman's side of the family tree just as well, however.

Let's start by redefining the man's mother in law to be the man's wife's mother. This approach helps us see each important step on the family tree. We'll also start by assuming the man's wife's mother to be the daughter in the mother-and-daughter relationship between the two mothers-in-law.

Given that, then we've worked up to the man's wife's mother's mother (or, the wife's maternal grandmother). For this top mother to be the woman's mother-in-law, the woman would have to be her son's wife. Putting this all together, we're looking for the man's wife's mother's mother's son's wife (*PHEW!*). If you don't understand that so far, go back and read it again until you do.

So, in Wolfram|Alpha, we enter wife's mother's mother's son's wife, to see how the family tree appears. In the tree that's drawn, “self” represents the man and “wifewife” represents the woman.

Unfortunately, Wolfram|Alpha doesn't display the name of this relationship, but it is easy to see from the diagram that the woman could be the man's wife's aunt. Wolfram|Alpha also says that this is the only possibile interpretation of this relationship, so we're done with this assumption.

To get the remaining solutions, we need to work through the man's wife's mother as the mother in the vaguely-defined mother-daughter relationship.

So now, we work up to the man's wife's mother's daughter. For this daughter to be the woman's mother-in-law, this daughter's son's wife has to be the woman. Putting all that together, we're looking for the man's wife's mother's daughter's son's wife.

At this point, we enter wife's mother's daughter's son's wife into Wolfram|Alpha to get a better idea of the relationship. We see from the diagram that the woman could be the man's wife's nephew's wife, as well.

Note that Wolfram|Alpha also mentions there are 3 possible interpretations to this relationship. Perhaps this is where we need to be looking for the tricky part that was mentioned earlier!

Consider that the man's wife's mother's daughter could not only describe the man's wife's sister, but the man's wife, as well! Let's run through the whole thing with this in mind.

This time, the man's wife's mother's daughter will simply refer to the man's wife. For this wife to be a mother-in-law of the woman, the woman must be her son's wife. That makes the woman the man's wife's son's wife.

The Wolfram|Alpha map of this relationship show that the woman is this case is simply the man's daughter-in-law. It also says that there are 2 interpretations of this relationship. What's the other one? If the man's wife had her son by another man, then the daughter is the man's step daughter-in-law.

So, at this point, we've actually found 4 answers to this puzzle:

• The woman is the man's wife's aunt.
• The woman is the man's wife's nephew.
• The woman is the man's daughter-in-law.
• The woman is the man's step daughter-in-law.

Under the assumptions I put forth at the beginning of this article, these are the only 4 possibilities. The answers written up in the original post agree with this, but answer #3 is worded oddly. It says, “the woman can be the wife of the man's nephew,” whereas it would be clearer to say, “the woman can be the man's wife's nephew.”

If you think you understand the process and wish to work through the problem with different assumptions, and/or want to work through the problem starting from the woman, rather than the man, I'd love to hear about your logic and results in the comments below!