I’ve just finished reading a book about the evolution of candidate interview questions in human resources – “Are you smart enough to work at Google?” by William Poundstone. It was fascinating, notwithstanding that it left me embarrassed and depressed at the number of sample questions I could not crack.
Once I overcame the idea that I would never have the maths skills to work at Google, I was mesmerised by one of the “Fermi” style questions in the book. Simply put: “How many golf balls can you fit in a Mini?”
A variant of the question was first posed in the late 1500s by Sir Walter Raleigh to the mathematician Thomas Elliott, in relation to stacking cannon balls on a ship. The astronomer Johannes Kepler – a friend of Elliott’s – argued (but could not prove) that the densest way of packing spheres was the method already widely used for cannonballs. Lay down the first “level” of spheres in a hexagonal array; then lay down a new level on top of that, fitting each new sphere in the depressions between three spheres on the preceding level (see diagram below).
In a large crate, each sphere is surrounded by 12 other spheres, and the “density” of this arrangement approaches 74%. Density in this context attributes a score of 100% to the sphere and nil to the space between spheres. The density of this hexagonal lattice compares to (i) 52% if the spheres are packed as if each were contained in a cube whose length equalled the diameter of the sphere; (ii) 55-64% if the spheres are packed randomly (the higher end of the range reflects post-packing agitation).
Carl Friedrich Gauss supplied a proof in 1831 that Kepler’s “hexagonal close packing” method was the densest among all possible regular lattice packing methods, but not among irregular arrangements. This remained a riddle until 1998, when Thomas Callister Hales announced a “proof by exhaustion” (or brute force), involving the checking of many individual cases (proven to be finite) using complex modelling. Referees have said that they are “99% certain” of the correctness of Hales’ proof, but it is not a proof in the purest sense of the term.
It got me thinking … we regularly ponder a number of important questions relating to strategy, human behaviour, process optimisation, capital allocation and risk. Some of these questions are characterised by significant tipping points, where value is either created or destroyed when a phenomenon varies over a narrow band of values. Important estimations of value, risk or change sit behind our decisions on these matters.
So how do we (and how should we) ponder the (equivalent of) the “density of the golf balls” issue that lies at the heart of the big decisions we make? Among other things:
- Are we blind to other issues in our pursuit of a narrow optimisation function? … How much oxygen does the driver of the Mini (no doubt a sad clown) need?
- Are we being tempted by inductive reasoning? In these scenarios, do we shut off our thinking to the potential cases where our theory no longer holds? How badly can we get it wrong, and what are the consequences if this happens? … At 74% density, will the clown and the Mini have enough inertia at 85mph to break through the front doors of the bank? What if the windscreen shatters and releases the balls?
- Is the search for a purist deductive solution plagued by diminishing marginal returns? What are the opportunities we miss as a result? … Above 74%, what is the implication (or value) of getting more golf balls in the Mini? How long do we have before the bank employees transfer the day’s cash from the drawers to the safe?
For me, these riddles are a great reminder of the importance of developing diversity of thinking in our team. Blind spots, temptation and stubbornness are always a risk, but diversity at least increases the chances of recognising these weaknesses.
I’m looking forward to bringing a couple of clowns and some golf balls to my next team meeting. I just hope that we don’t identify a riddle that lasts half a millennium.