Writing

The rice and the chessboard

We talk constantly about AI improving exponentially, and everyone nods along — but almost no one really feels it: our minds reason in sums, not doublings. A very old story, with a chessboard and a handful of rice, is still the best way to grasp what we’re missing.

Daniele Bianchini · June 2026 · English · ~6 minutes

A short explainer, written with the help of Claude Opus 4.8 (June 2026). Leggilo in italiano →

“Artificial intelligence is improving exponentially.” You hear it everywhere, and almost everyone nods. The trouble is that “exponential” has become a filler word, a vague synonym for “grows a lot.” The actual concept is precise and deeply counterintuitive, and missing it means getting the direction of things systematically wrong. I want to make it obvious without a single formula, using the oldest example there is.

The legend

The story, in one of its many versions, is set at the court of a Persian king. The inventor of chess presents the game to the ruler, who is enchanted and offers any reward he likes. The inventor asks for something seemingly modest: one grain of rice on the first square of the board, two on the second, four on the third, and so on, doubling all the way to the sixty-fourth. The king, almost offended by so humble a request, agrees on the spot. He has just made a mistake that will cost him his kingdom.

The first squares look like nothing

Walk the first row. One, two, four, eight, sixteen, thirty-two, sixty-four, a hundred and twenty-eight: by the end of the first row, on square 8, there are 128 grains — barely a spoonful. By square 16 we’re at about thirty thousand, a bowl. By square 21 we pass a million, a sack. At the halfway point, square 32, that single square holds over two billion grains, and the running total comes to nearly four and a half billion — a large planted field. An enormous number, but still somehow imaginable. Then the second half begins, and language stops keeping up.

The turning point

Square 33, on its own, holds more grains than the entire first half of the board combined. It is worth reading twice, because it is the whole point: a single square, the thirty-third, weighs more than the thirty-two before it put together. And every square after it does the same to all the ones that came before. By square 64 the grand total is 2⁶⁴ − 1 grains: more than eighteen quintillion. By weight, that is several hundred billion tonnes of rice — many centuries of the entire world’s harvest, heaped onto one chessboard. In the various endings of the legend the inventor is made prime minister, or beheaded; either way, the king has learned the hard way what a doubling is.

The doubling, square by square:

The right-hand column says the important thing: the total up to any square always equals, give or take a single grain, the grains sitting on the next square. The entire past, summed up, is worth just the one step that comes after it.

The two lessons our minds refuse

This little story holds the two facts about the exponential that our intuition keeps pushing away.

• Every step is worth more than the whole road behind it. On a linear path, the next step looks like the last one. On an exponential path, the next step is bigger than everything you have already covered. That is why “it only got a little better this year” can be true and misleading at once: that small visible increment can exceed the sum of all previous progress.
• The action is all in the second half. The first half is boring, and that is precisely the trap. For thirty-two squares it looks like a modest, almost linear handful. All the drama is loaded into the end. Kurzweil calls it “the second half of the chessboard” [3]: the stretch where an exponential, after a very long unremarkable run-up, starts producing numbers that no longer fit inside the world.

Why our minds get it wrong

We are not stupid, nor lazy with numbers: we are built for the linear. For almost all of human history, effort and result were roughly proportional — ten steps, ten metres; double the field, double the harvest. Nothing in daily life doubles at regular intervals. So when something does, we read its early part, slow and gentle, as if it were the whole curve, and we draw a straight line into the future. It is not an error of arithmetic, it is an error of shape: we are using the wrong silhouette.

And now, AI

The thing that, in AI, doubles at regular intervals is compute: the raw operations used to train the leading models. By Epoch AI’s reckoning it has doubled roughly every six months for more than a decade [1] — about four times a year, where Moore’s law took two years for a single doubling. Compute is not intelligence, and I’ll add the necessary caveats in a moment, but it is the closest thing AI has to “grains per square,” and it is well into the second half of its board.

This is why predictions keep failing in the same direction. People look at what a model can do today and draw the usual straight line: a bit better next year, a bit better the year after. But if the input doubles every six months, “next year” is about four times this year, and three or four years out is a hundredfold — more change ahead than behind. The people who said “AI will never” pass a bar exam, write working code, or hold a real conversation weren’t foolish: they drew a straight line on an exponential curve, and the board did what the board always does.

For years it genuinely looked like a toy. That long boring first half — decades of academic curiosity — is exactly what the chessboard predicts. The toy phase isn’t evidence that nothing will happen: on an exponential, it is the stretch right before everything does.

The caveats, because “exponential” is also the salesman’s word

I want to be precise here, because “exponential” is also the favourite term of anyone with something to sell. Two honest qualifications. First: compute is an ingredient, not the result. More operations buy more capability, but with diminishing returns, not one for one; doubling the compute does not double “intelligence” in any clean sense of the word. Second: no real exponential lasts forever. In nature they are almost always the early part of an S-curve — bacteria in a dish, a virus in a population — that eventually bends when it hits a limit: data, energy, money, physics.

What matters, though, is which way humans get it wrong, systematically. Faced with an exponential, we almost never overestimate its early part: we underestimate it, badly, every time, because we keep reaching for the straight line. Even if AI’s curve were to bend tomorrow, the safe bet of the past decade — and probably of the next few years — has been to assume we are underestimating, not overestimating. The chessboard doesn’t tell you where the board ends; it tells you to stop drawing straight lines across it.

The useful thing to do

So the next time someone says AI is improving exponentially, the useful reaction isn’t excitement, and it isn’t alarm either. It is to remember the rice and the chessboard, and to notice that your gut, just then, quietly drew a straight line — and to correct it. That correction, more than any precise prediction, is what it means to take the exponential seriously.

The numbers

• The board total: 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains. At roughly 25 mg each, that is around 460 billion tonnes; against a world rice output of about 520 million tonnes a year, it is nearly a millennium of the entire planet’s harvests. The figure swings quite a bit with the weight of a single grain (between 20 and 30 mg), but the order of magnitude doesn’t.
• The compute doubling: about 6 months is Epoch AI’s estimate for recent models [1]; between 2012 and 2018 OpenAI measured an even faster pace, roughly every 3.4 months [2]. Moore’s law, by comparison, doubles about every two years.

Sources

1. Epoch AI, trends in training compute for machine-learning models (doubling roughly every 6 months). epoch.ai
2. OpenAI (2018), “AI and Compute”: doubling roughly every 3.4 months between 2012 and 2018. openai.com
3. R. Kurzweil, the “second half of the chessboard” metaphor (from The Age of Spiritual Machines, 1999). For the legend and the math, “Wheat and chessboard problem.” wikipedia.org