Sissa's chessboard: how many grains of rice? The calculation explained step by step

The legend is told in many ways, but its mathematical mainspring never changes. A king wants to reward the inventor of chess, a sage whom tradition calls Sissa. The sage asks for a reward that looks modest: one grain on the first square of the chessboard, two on the second, four on the third, and so on, doubling on each square up to the 64^th. The king agrees with a laugh… then discovers that all the grain in the kingdom will not be enough.

The legend of Sissa's chessboard in a few lines

Depending on the version, the reward is in grains of wheat or grains of rice, and the king goes by different names (Belkib, for example, in several French-language accounts). The story is a legend: it does not tell the real origin of chess. The oldest known written account of this story is attributed to the historian Ibn Khallikan, in 1256. What matters here is the calculation, and it is relentless.

Sissa's chessboard problem

A problem that consists of placing 1 grain on the first square of a chessboard, then doubling the number of grains on each following square. Square n carries 2ⁿ⁻¹ grains, and the total over the 64 squares is 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains.

The typical wording of a 4e exercise

In one form or another, here is the exercise found in many secondary school workbooks (4e in France, pupils aged 13–14):

The calculation step by step

1. Grains on a given square

On the 1^st square there is 1 grain, which can be written 2⁰. On the 2^nd, 2 = 2¹. On the 3^rd, 4 = 2². On the 4^th, 8 = 2³. At each square the exponent goes up by 1, but it is always one behind the square's number. So square number n carries 2ⁿ⁻¹ grains: the 10^th square carries 2⁹ = 512, and the 64^th carries 2⁶³ = 9,223,372,036,854,775,808.

2. The total: one observation that does it all

Let us add up the first squares: 1 + 2 + 4 + 8 + 16 = 31. The next square carries 32. The total of the previous squares is always the next square minus 1. This can be checked at every step: 1 = 2 − 1, 1 + 2 = 4 − 1, 1 + 2 + 4 = 8 − 1… The total of the 64 squares is therefore the “65^th square” minus 1, that is 2⁶⁴ − 1.

3. The proof, for older pupils

Let S = 1 + 2 + 2² + … + 2⁶³. Multiply by 2: 2S = 2 + 2² + … + 2⁶³ + 2⁶⁴. Subtract the first line from the second: almost all the terms vanish, leaving 2S − S = 2⁶⁴ − 1. So S = 2⁶⁴ − 1. This is a special case of the sum of a geometric sequence with common ratio 2, studied in the première year (age 16–17).

2⁶⁴ − 1 = 18 446 744 073 709 551 615

In words: eighteen quintillion, four hundred and forty-six quadrillion, seven hundred and forty-four trillion, seventy-three billion, seven hundred and nine million, five hundred and fifty-one thousand, six hundred and fifteen. A school calculator generally displays 2⁶⁴ as 1.844674407 × 10¹⁹: it does not have enough digits to write it out in full, so it cannot show the difference between 2⁶⁴ and 2⁶⁴ − 1.

Square-by-square table

Two remarks strike pupils. First, the first rank of the chessboard (8 squares) holds only 255 grains: a small handful. Second, the last square on its own holds one more grain than the 63 previous squares put together. This is the hallmark of exponential growth: each doubling weighs as much as all the history before it.

How many tonnes of rice? An order of magnitude

To go from grains to tonnes, you need an assumption about the mass of a grain, which varies by variety. A grain of rice weighs a few hundredths of a gram: the Wikipedia article on the problem uses 0.04 g, and agronomic measurements on some varieties give slightly more than 0.025 g (about 26 g per thousand grains). The two assumptions give:

Calculation: 18,446,744,073,709,551,615 × mass of one grain. Compared with a world harvest of about 776 million tonnes of rice in 2022 (FAOSTAT data as reported by Wikipedia). With wheat grains of 65 mg, the assumption used by the English-language Wikipedia article, the total exceeds 1,199 billion tonnes.

Another way of putting it: counting one grain per second, without ever stopping, it would take about 585 billion years to count all the grains, more than forty times the estimated age of the Universe.

Finding the order of magnitude without a calculator

A useful trick in class: 2¹⁰ = 1,024, which is roughly 1,000 = 10³. Now 2⁶⁴ = 2⁴ × (2¹⁰)⁶, so roughly 16 × (10³)⁶ = 16 × 10¹⁸ = 1.6 × 10¹⁹. The true value is 1.84 × 10¹⁹: the approximation underestimates a little, because each “1,024 ≈ 1,000” loses 2.4%, six times in a row. It is a good exercise on powers and on scientific notation.

The questions pupils ask

What if we triple instead of doubling?

With tripling, square n carries 3ⁿ⁻¹ grains, and the total is (3⁶⁴ − 1) ÷ 2. The same method (multiply S by 3, then subtract) gives the result. The number obtained exceeds 10³⁰.

How many squares are needed to exceed a million grains?

The total after n squares is 2ⁿ − 1. Since 2¹⁹ = 524,288 and 2²⁰ = 1,048,576, you need 20 squares to exceed a million grains in total. And you need 30 squares to exceed a billion (2³⁰ − 1 = 1,073,741,823).

Why does 2⁶⁴ keep coming up in computing?

Because a number written on 64 bits can take exactly 2⁶⁴ different values. The largest positive integer that can be encoded in unsigned 64 bits is therefore 2⁶⁴ − 1: the same number as Sissa's.

Going further

The chessboard hides other gigantic numbers. The number of possible chess games, estimated by Claude Shannon, makes Sissa's look tiny: see more very large chessboard numbers with the Shannon number. For more geometric problems, head to the chessboard's mathematical puzzles. And to keep your mind sharp between two exercises, Le Palais des Échecs offers a mathematical logic test to practise with.