The Combinatorics of Family Chaos

For those of you who read this blog regularly, you’ve likely noticed a lull in my posting.  That’s because, about two weeks ago, my wife & I welcomed a new addition to the family.  Given that the most common response to our decision to add a fourth child to the family has largely been “borderline insanity”, I felt it was appropriate to share some of my thinking on the complexity that comes with every new addition.

The Wrong Model: Linear

When a couple decides to have a second child, you are quickly inundated with advice on how to manage the complexity.  The most common refrain you hear is: “Don’t worry, you can still field man-on-man coverage.”  Another popular version of this advice is: “At least you’re not outnumbered.”

The implication here is that managing the family is fundamentally a relationship between parents & kids, like this:

Parental Ratio = # of Parents / # of Kids

With the implication that somehow, as long as the parental ratio is greater than or equal to one, you’ll be able to manage.

Unfortunately, I’ve found that this description of complexity dramatically understates the drama of real family life.

The Right Model: Combinatorics

Instead 0f focusing specifically on the number and types of nodes in the family graph, I think it’s more useful to think about the nature of emotional entanglements (aka “drama”) and understand that they tend to require at least two people, but can easily involve more.  As a result, the complexity of family life can be more accurately modeled as the number of two-party relationships in a family that can engage in drama.

Initially, a couple has exactly one potential pair:

  • Adult 1 <-> Adult 2

However, once you add a single child to the mix, you immediately add two more vectors of potential drama:

  • Adult 1 <-> Child 1
  • Adult 2 <-> Child 1

It’s worth noting that it’s sometimes unclear whether a three-party argument is truly a single argument or actually a combination of two or three two-party arguments, but let’s just roll with the simplified assumption for now that all drama can be decomposed to pair-based drama.

Pascal’s Triangle actually makes calculating this number for any size family trivial.  This means that:

  • Two family members (0 kids): 1 drama pair
  • Three family members (1 kids): 3 drama pairs
  • Four family members (2 kids): 6 drama pairs
  • Five family members (3 kids): 10 drama pairs
  • Six family members (4 kids): 15 drama pairs

It’s combinatoric, specifically in the form of:

family complexity  = # of family members  choose 2

Which is a fancy way of saying each new child adds a new relationship to the mix for every existing family member.  This sequence is also known as the triangular numbers.

For those of you who have or come from large families, let me know if this lightweight graph theory matches your experience.

Julia Elizabeth Nash

Julia Elizabeth Nash
Born 10.0 lbs, 21 inches
Single handedly increased Nash family complexity by 50%

9 thoughts on “The Combinatorics of Family Chaos

  1. The interesting dynamic that you don’t note in your analysis is the potential that someone could alleviates the “emotional entanglements” i.e. drama rather than add to them. Julia might actually be someone who ameliorates ‘situations’ in personal dynamics.

    Either way, you’ve got lots of entertainment at home. Enjoy! and congratulations on your new family addition.

  2. Adam, congrats on the addition of your beautiful new daughter! As the eldest of four children from a family of six, I can definitely attest to the increase in complexity. Wishing you and your family the very best.

  3. I enjoyed your analysis immensely. If you add chemistry to combinations, I think you might have the full model. Could start with thinking of each family member as an element. Who reacts with whom? Any inert gases? Any radioactive? How much pressure is the system at? Wait until Julia is at least 5 before adding any of this complexity. Keeping it simple now will help you enjoy it more. Very happy for you all!

  4. Pingback: Did You Like Being an Executive in Residence (EIR) | Psychohistory

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