There is an old story, told in slightly different versions across India, Persia, and medieval Europe. A king asks a wise man — sometimes a court astronomer, sometimes the inventor of chess — to name any reward he likes for his cleverness. The wise man, with a smile, asks for something apparently modest.
"Only grains of rice, Your Majesty. Place one grain on the first square of a chessboard. Two on the second. Four on the third. Eight on the fourth. Keep doubling. On each square, twice as many grains as on the one before. When the board is filled, give me whatever rice has accumulated."
The king, amused, agrees. The board has 64 squares. The king expects a modest sack of rice.
The answer is that the kingdom's entire granary would not contain it. The answer, in fact, is that the rice owed to the wise man dwarfs all the rice ever harvested in human history.
The picture
Running the numbers
Square 1 has 2^0 = 1 grain. Square 2 has 2^1 = 2 grains. Square 3 has 2^2 = 4 grains. In general, square k has 2^{k-1} grains.
A handful of rice in India has roughly 50 grains. Let's walk up the chessboard and see where we stand.
- Square 10 (row 2 corner): 2^9 = 512 grains. About ten handfuls. A small bowl.
- Square 20: 2^{19} \approx 524{,}000 grains. About 12 kg. One large sack.
- Square 30: 2^{29} \approx 537 million grains. About 13 tonnes. One truckload.
- Square 40: 2^{39} \approx 5.5 \times 10^{11} grains. About 13{,}000 tonnes. The cargo of a large ship.
- Square 50: 2^{49} \approx 5.6 \times 10^{14} grains. About 14 million tonnes. Roughly one year's national rice output of India.
- Square 60: 2^{59} \approx 5.8 \times 10^{17} grains. About 14 billion tonnes. More than all rice ever grown by humankind.
- Square 64: 2^{63} \approx 9.2 \times 10^{18} grains. About 230 billion tonnes — a pile that, if heaped in one place, would tower above Mount Everest.
And that is just the grains on the final square. The total rice on all 64 squares is
In tonnes, about 460 billion. To put that in context, annual global rice production is about 500 million tonnes. At that rate, producing the promised reward would take roughly a thousand years of humanity's entire rice crop.
Why the total is 2^{64} - 1: the sum 1 + 2 + 4 + \cdots + 2^{n-1} is a geometric series. A neat way to see its value: the next power, 2^n, equals 1 + 1 + 2 + 4 + \cdots + 2^{n-1} (binary 100\ldots0 equals one more than 11\ldots1 with n ones). So 2^n = 1 + (\text{sum}), giving \text{sum} = 2^n - 1. In our case, 2^{64} - 1.
What the story is really about
The king was not stingy. He was tricked by a mathematical intuition that all humans share: we feel increments as additive, so we unconsciously picture the chessboard as 64 squares with "some more rice" on each. Our intuition wants to say "maybe fifty times the rice on the first square, or a hundred times — a manageable amount."
What our intuition misses is the compounding: each doubling is a whole new base from which the next doubling operates. By square 10, you have 512 grains — a small bowl. Fine. But the first square contributed 1 grain; the tenth contributes 512. The ratio is 512 : 1, and the next doubling is no longer 1 \to 2 but 512 \to 1024. The rice on each new square always matches the total rice on all the previous squares combined (almost exactly — off by one grain).
By square 64, the increment from square 63 to square 64 is itself more than all the rice on squares 1 through 63 put together. This is the geometric hallmark of exponential growth: the next step is as big as everything that came before. No linear or polynomial rule can keep up. Eventually, exponential growth outpaces anything that is not itself exponential.
Where this shows up
The chessboard is a parable, not an actual event, but the same geometry governs several real situations that every student should recognise.
- Cellular biology. One bacterial cell dividing every 20 minutes would, in 24 hours, fill a space larger than Earth — if it had unlimited food and space. Bacteria don't usually do this, because the environment runs out. But the unchecked rate is exponential.
- Compound interest. Money growing at 7\% per year doubles in roughly 10 years (rule of 72). ₹1 lakh becomes ₹2 lakh in ten years, ₹4 lakh in twenty, ₹1.3 crore in forty. This is a slower doubling than rice, but it is still a doubling, and given enough squares (enough decades), the pile runs away.
- Moore's law of computing. Transistor counts on microprocessor chips doubled roughly every two years for fifty years. A chip from 1980 had tens of thousands of transistors. A modern chip has tens of billions — a 10^6-fold increase in forty years. That is "twenty doublings," and each one was about as impactful as the previous twenty combined.
- Pandemic spread (without intervention). A disease with R_0 = 2 means each infected person infects two more. After 15 generations of infection, that is 2^{15} = 32{,}768 cases from one. This is why doubling time is the metric public-health officials watch.
In every one of these cases, the chessboard story is the right thing to keep in mind. The first few squares are unimpressive. The late squares are catastrophic. The gap between intuition and reality is the story's reward.
The formula
The parable can be stated in one line of exponent law, which is the reason it lives in this chapter:
Every time you meet a doubling process — in biology, economics, computing — the pattern 2^{k-1} is the skeleton. The chessboard is just the clearest picture of it.
Carry-away
- Doubling 64 times gives 2^{64} \approx 1.8 \times 10^{19} — a number with twenty digits.
- The next term of a doubling sequence always exceeds the sum of all previous terms.
- Exponential growth outruns intuition, so build the reflex to compute doubling times (rule of 72) before reasoning about compound interest or population growth.
- Powers of 2 from 2^0 to 2^{10} are worth memorising: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. After that, every further decade multiplies by about 10^3, so 2^{20} \approx 10^6, 2^{30} \approx 10^9, 2^{40} \approx 10^{12}.
Related: Exponents and Powers · Exponent Slider: Watch 2^x Sweep Through 1/8, 1/4, 1/2, 1, 2, 4, 8 · Percentages and Ratios · Number Systems