How Much Rice Is on the Chessboard?

The first squares

Start small. The first square has 1 grain. Each square after it has twice as many as the one before.

Look at the two rows of your table. Two patterns are hiding there.

Why one more?

Here is a picture of the pattern. Each grain is drawn as one block, so you can count them.

Now here is why it must happen, every time, all the way to square 64. Put one extra grain beside square 1. Square 1 and the extra grain make 2, exactly what square 2 holds. Add square 2: 2 + 2 makes 4, exactly what square 3 holds. Add square 3: 4 + 4 makes 8, which is square 4. At every step you add a square to a pile that is already as big as that square, so the pile doubles and becomes the next square. The pile is always “all the squares so far, plus one”, and it is always exactly the next square. Take the extra grain away again, and the squares so far are one less than the next square.

Doubling, written with powers

Multiplying by 2 again and again gives the powers of 2: 2, 2 × 2 = 4, 2 × 2 × 2 = 8, and so on. You know the short way to write them from Grade 8’s indices: 2 × 2 × 2 = 2³, and 2 × 2 × … × 2 with ten 2s is 2¹⁰ = 1,024.

Square 1 has had no doublings yet, square 2 one, square 3 two. So square n has had n − 1 doublings:

$$\text{square } n \text{ holds } 2^{n-1} \text{ grains.}$$

Square 1 holds 2⁰ = 1 grain, which is Grade 8’s rule that any number to the power 0 is 1. Square 64 holds 2⁶³ grains. And since the squares so far are always one less than the next square, the total on the first n squares is

$$1 + 2 + 4 + \cdots + 2^{n-1} = 2^{n} - 1.$$

So the whole board holds 2⁶⁴ − 1 grains. The board on the page writes the bigger squares this way, 2¹³ for square 14, once their numbers no longer fit.

Growth like this, where something is multiplied by the same number at every step, is called exponential growth: the exponent, the small raised number, goes up by one at each step. Doubling is exponential growth by 2. Growth where the same amount is added at each step (1, 2, 3, 4… or Rs 1,000 more every day) is linear growth: on a graph it is a straight line.

A quick estimate

You can estimate 2⁶⁴ in your head with one trick: 2¹⁰ = 1,024, which is very close to 1,000 = 10³. Then

2⁶⁴ = 2⁴ × 2¹⁰ × 2¹⁰ × 2¹⁰ × 2¹⁰ × 2¹⁰ × 2¹⁰ ≈ 16 × 10³ × 10³ × 10³ × 10³ × 10³ × 10³ = 16 × 10¹⁸.

The exact number of grains on the whole board is

2⁶⁴ − 1 = 18,446,744,073,709,551,615

a number with 20 digits. The estimate, 16 × 10¹⁸, was a little low: each 1,024 is 2.4% more than 1,000, and the six of them together make the true number about 15% bigger than the estimate. Still, it was right to the nearest power of ten, from mental arithmetic alone.

Writing the number

Nepal groups the digits of a big number in a different way from most of the world: the first comma after three digits, then a comma after every two, so that the groups are thousand, lakh, crore, arab, kharab. The same number of grains, written both ways:

Written	The board’s grains
Nepal’s way	1,84,46,74,40,73,70,95,51,615
The international way	18,446,744,073,709,551,615
In scientific notation (Grade 8)	about 1.84 × 10¹⁹

Above kharab, the names are rarely used, and lists don’t always agree. The Nepali Brihat Shabdakosh gives महाखर्ब (mahakharab, 100 kharab, 10¹³), पद्म (padma, 10¹⁵), शङ्ख (shankha, 100 padma, 10¹⁷) and महाशङ्ख (mahashankha, 100 shankha, 10¹⁹). With those, the board’s grains are read:

184 shankha, 46 padma, 74 mahakharab, 40 kharab, 73 arab, 70 crore, 95 lakh, 51 thousand, 615.

In international names it is 18 quintillion, 446 quadrillion, 744 trillion, 73 billion, 709 million, 551 thousand, 615. Topic 2, How Long Is a Crore Seconds?, is all about getting a feel for such names.

Weighing the board

At about 20 mg a grain, the whole board weighs

18,446,744,073,709,551,615 × 20 mg ≈ 369 billion tonnes (3.69 × 10¹¹ tonnes).

To compare	How much rice	The board is about
Nepal’s paddy harvest in 2081/82 (2024/25)	5.95 million tonnes	62,000 years of it
The world’s rice harvest forecast for 2026/27	552.5 million tonnes	668 years of it

The last square alone holds half of all of it: 2⁶³ grains, about 184 billion tonnes. To pay, the king would have needed every grain of rice the whole world grows, every year, for 668 years.

Here is a choice a moneylender might offer you. Take one:

• Adding: Rs 1,000 on day 1, and Rs 1,000 more on every day of the month.
• Doubling: 1 paisa on day 1 (a hundredth of a rupee), 2 paisa on day 2, 4 paisa on day 3, and on every day twice the day before’s gift, for 30 days.

The race shows the two things everyone learns about exponential growth:

• It starts slower. For two weeks, doubling from 1 paisa is nothing next to Rs 1,000 a day: its bars can’t even be seen.
• It always wins in the end. Adding brings the same Rs 1,000 every day, but doubling’s daily gift doubles, so sooner or later it passes Rs 1,000 (on day 18, Rs 1,310.72) and keeps on doubling. Its total soon catches up.

Making the adding offer ten times bigger doesn’t save it. Rs 10,000 a day is caught on day 25 instead of 22; Rs 1 lakh a day on day 29. A thousand times more money a day only buys about ten days, because ten doublings multiply by about a thousand.

Fold a sheet of paper in half, and it is two layers thick. Fold it again: 4 layers. Again: 8. Every fold doubles the layers, just as every square doubles the rice. So after n folds the paper is 2ⁿ layers thick. A sheet of ordinary copy paper is about 0.1 mm thick. How many folds to reach the Moon?

On a ladder where each rung is 100 times the rung below, every fold is the same small step: doubling is a steady climb, never a sudden jump. Fourteen folds make a stack about your height. Twenty-seven pass Mount Everest. And 42 folds make a stack 439,805 km tall, past the Moon, which is on average 384,400 km away. Nine more, 51 in all, and the stack would pass the Sun.

But try it.

Halving is doubling backwards, and it gets small just as fast. Halve 1 kg of rice (50,000 grains), then halve the half, and so on: after only 16 halvings you are down to less than a single grain.

Doubling numbers run away so fast that nothing real can keep doubling for long. Bacteria show it best. Under ideal conditions in the laboratory (warm, with plenty of food), the bacterium E. coli can split in two about every 20 minutes. That is three doublings an hour. One bacterium becomes 8 after an hour, 4,096 after 4 hours, more than a crore after 8 hours (2²⁴), and more than a billion after 10 hours (2³⁰). After a day, 72 doublings, there would be more bacteria than grains of rice on the whole chessboard.

It doesn’t go on. Long before that, the bacteria run out of food and room, and the doubling slows down and stops. Real growth often starts out exponential and then levels off. So an exponential pattern is a good guide while there is room to grow, and a bad one when you use it far past that: always ask what will run out.

The Grade 10 book adds the whole board up as a sum of a geometric series, with a formula for any multiplier: 6.4 Sum of a geometric series.

• How Long Is a Crore Seconds? A feel for lakh, crore, million and billion, and how far apart they are.
• Can You Add Halves Forever and Still Get One? Halving the other way: ½ + ¼ + ⅛ + … never passes 1.
• What Is e, and Why Does Growth Keep Finding It? (Grades 9–12) Growth that doubles not in steps but every moment.
• What Is the Biggest Number Anyone Has Used? (Grades 9–12) Numbers that make 2⁶⁴ look tiny.
• How Many Moves to Move 64 Disks? (Computing, Going Further) The Tower of Hanoi: its fewest moves are the same 2⁶⁴ − 1, and you program it.
• Can a Computer Check Every Possible Chess Game? (Computing, Going Further) Where even doubling is too slow a word. (Coming soon.)
• Grade 10, Sum of a geometric series: the board added up by a formula, for doubling, tripling or any multiplier.
• Grade 8 Computing, From Blocks to Python: Python works out the whole board’s rice exactly, all 20 digits.