In the story, the royal dupe who agrees to this deal ends up owing the sage far more rice than exists on the planet. The fact that this isn’t obvious reflects our mental inability to cope with the impact of exponential doubling. There are 64 squares on a chessboard, and we start with just a few grains on each square. So, however rational we are, it’s hard to get away from thinking that the outcome must be something comparable with 64 times a sizeable but manageable number. Perhaps around 64 million or 64 trillion. The reality, however, is very different.
Let’s take a look at the total number of grains. As we have seen, on the first square we put one grain. With two squares there’s one grain on the first square and two on the second – three in total. With three squares there are seven grains. And with four squares there are 15. Nothing frightening so far. That sequence of numbers – 1, 3, 7, 15 … – is just one short in each case from a more familiar series: 2, 4, 8, 16 … The powers of the number two. This means we can quickly calculate how many grains there are with n squares occupied this way as 2n–1. We multiply two by itself n times and take away one. Here we can see very clearly how that exponential part is creeping into the calculation.
So, the total amount of rice required to fill up the whole board would be 264–1. Written like that, it still doesn’t look too scary, as it’s only little old 2 that is being multiplied by itself. But to put it another way, that is 18.5 billion billion (if you want to be precise, it’s 18,446,744,073,709, 551,615 grains). A whole lot of rice. Very roughly it’s about 300 billion tonnes of the stuff – the amount the world would currently consume in 600 years.
The sheer volume of rice involved is fascinating – but what has this to do with genealogy? Exponential doubling also comes into a family tree due to a simple fact that we’re all aware of, even though we tend not to think through the consequences. Each individual person on a family tree – you, for instance – will have exactly two biological parents. This means that if we ignore siblings (we will come back to them) and simply trace back an individual’s tree into history, the number of people in each preceding generation doubles. In your tree you will have two parents, four grandparents, eight great grandparents and so on.‡
Linear versus exponential growth: the linear line goes up one each time, the exponential line doubles each time.
Just like the chessboard rice grains, these numbers do not initially seem to be very extreme, until we start to combine them with a realistic count of generations going back in time. For convenience, generations used to be treated as 25 years, because it’s easy to do the maths and it was a reasonable approximation for the average age at which people became parents. These days, 30 is more like a sensible average, but for most of history the value tended to be lower, so 25 may be the best number. We’ll try both.
Just like the chessboard, the count of your directly linked ancestors goes through exponential doubling – in this case doubling with every generation, rather than with every square on the board. There’s one of you, three people with your two parents, seven people with your four grandparents and so on – that familiar series of 1, 3, 7, 15… So, looking back n generations in your past, we get to a total of 2n–1 people, in a tree stretching back into history from you as the root. And this is true for every one of the near 8 billion people alive today (which brings in siblings). Each individual has a tree stretching back in history the same way.
THE MISSING BILLIONS
Now for the big reveal. It has been estimated that around 110 billion people have lived since Homo sapiens came into existence a couple of hundred thousand years ago. That’s obviously a guestimate, but not a bad one. The figure that is most often quoted is 108 billion, based on a calculation by a group called PRB. I’ve made it 110 billion, partly because 108 billion gives a spurious feeling of accuracy and partly because one of their assumptions, that Homo sapiens has been around for 50,000 years, is a significant underestimate based on current data.
So, how many generations does that 110 billion represent? Just using your personal tree, we would need to go back around 37 generations. This is because 236 is around 68 billion and 237 is around 137 billion. If there were a totally separate tree for each person alive today, we would only need about 34 generations. Clearly that’s a reduction too far, as siblings will share the same tree – so it would be realistic to go for, say, 36 generations.
If we use 25-year generations that takes us back just 900 years into the past, or with 30-year generations we get back 1,080 years. Using this simplistic calculation, humans should have only been around for 900 to 1,000 years. In reality, though, we know that history goes back several thousand years further and archaeologically and palaeontologically speaking we can say that Homo sapiens has been in existence as a species for perhaps 200,000 years. Being generous and using the longer figure of 30 years for a generation, that’s 6,666 generations. Which would mean 26666 people in the single family tree starting from you, a phenomenally large number. It’s approximately 4 × 102006. To put that number into context, the number of atoms in the universe is often estimated to be around 1080.
Clearly, something has gone horribly wrong with this calculation. What these numbers reflect is that the nice well-ordered family tree we get from genealogists has been pruned incredibly tightly. We don’t have neat, spreading trees, but complex tangles. Go back a few generations and you will find that branches entwine and interlink in a more and more complex fashion. Increasingly large numbers of the inhabitants of each generation will be duplicated over and over again as the same historical person appears in different branches. And this effect will become stronger as you go further back in time because of low mobility. You don’t have to go back very far in history to get to a stage where the majority of people never ventured far from their home village. They would not have the whole world available to them as breeding partners, but rather a tiny gene pool.
Something had to give way to go from more than 4 × 102006 to a mere 110 billion (for a clearer comparison, that’s 1.1 × 1011). And that’s where we can all share Danny Dyer’s excitement. It’s not that there is anything wrong with his genealogy, but rather that we can say with certainty that everyone has royal ancestry, thanks to the application of statistical analysis to the numbers of branches and linkages in our ancestry. If we go back far enough in any such intertwined tree we end up with regional common ancestors – people who for those of a certain region we can guarantee will be in their family tree. This is true of not just a few of the potential ancestors, but vast numbers of them. Go back far enough, in fact, and we can say that you are related to every single person from your region who has living descendants. That will apply to kings and queens, just as much as minstrels and servants, murderers and vagabonds. They’re all there in your family tree.
EVOLVING A THEORY
Before we manage to resolve those vast numbers of missing ancestors, we need to get a feel for how far back we need to go. In Chapter 6 we will be looking at where humans came from, but here it’s just a case of knowing where to stop looking into the past with those family trees. And to find the right point to make the break, we are going to have to deal with the E word: evolution.