Summary of the tale

This legend tells how an Indian sage astonished a king with an apparently modest request: put one grain of wheat on the first square of a chessboard, then double it on each following square. Amused, the king accepted without hesitation! Yet this sequence of doublings hid a mathematical trap — a reward that could never be paid.

About the tale

This Indian legend features Sissa ibn Dahir, a mythical figure to whom the invention of Chaturanga — ancestor of chess — is attributed.
As the story goes in an old manuscript, the king asked Sissa what reward he wished for his invention.

The sage answered with the wheat-and-chessboard problem, showing that things are not always what they seem.

Simple to tell, with a clear progression, this story leads pupils to wonder… even before doing any calculations.

👉 Discover the tale

Maths

Powers, doubling, huge numbers…
and an overwhelmed king

When Sissa proposes to double the grains on each square of the chessboard, he is (silently) posing a mathematical challenge: exponential growth.

On the 64^th square the total reaches $2^{64}-1 = 18{,}446{,}744{,}073{,}709{,}551{,}615$ grains!

👦🏻 Target age: 9–10 (4th–5th grade)

⏰ Estimated duration: 1 hour

📎 Materials: calculator, board, student worksheet

🎯 Learning objectives

Develop the following skills:

• Compute and use powers of 2.
• Understand and sum the terms of a simple geometric sequence.
• Handle the notation and reading of very large numbers.
• Interpret a fast-growth situation.

🔢 Key mathematical ideas

• Powers of 2:

$2^0=1\quad;\quad 2^1=2\quad;\quad 2^2=4=2\times2\quad;\quad 2^3=2\times2\times2=8\quad;\quad$ etc.

In general: $2^n = 2\times 2^{\,n-1}$.

• Geometric sequence with ratio 2:

First term $u_0=1$ ;

$u_n = 2^n$ ;

Sum of the first $n+1$ terms:

$S_n = 2^{\,n+1}-1$.

• Examples of partial sums:

$S_0 = 1$ ;

$S_1 = 1+2 = 3$ ;

$S_2 = 1+2+4 = 7$ ;

$S_3 = 1+2+4+8 = 15$.

Notice that $S_1=2^2-1$, $S_2=2^3-1$, $S_3=2^4-1$, … Hence $S_{63}=2^{64}-1$.

• Large numbers (short scale, English):

million = $10^6$ ; billion = $10^9$ ; trillion = $10^{12}$ ; quadrillion = $10^{15}$ ; quintillion = $10^{18}$ ; sextillion = $10^{21}$ ; septillion = $10^{24}$.

🟢 Activity 1. Powers of 2

Compute $2^5$, $2^8$, $2^{10}$, $2^{12}$.

🟢 Activity 2. Sum of a geometric sequence

Verify that $1+2+4+\dots+2^5 = 2^6 - 1$.

🟢 Activity 3. Grains on the chessboard

How many grains are on the 10th square?

How many grains in total over the first 10 squares?

And over all 64 squares?

🟢 Activity 4. Reading and writing large numbers

Write in words:

$2^{16}=65{,}536$ ;

$S_{63}\approx 1.844\times 10^{19}$.

🟢 Activity 5. When growth explodes

Why does the sequence of grains become “astronomical”?

Give concrete examples of doubling: population growth, a water leak, folding a sheet of paper… How would you interpret these?

Answer key

1. Powers of 2

$2^5=32\ ;\ 2^8=256\ ;\ 2^{10}=1{,}024\ ;\ 2^{12}=4{,}096$.

2. Sum of a geometric sequence

$S_5 = 1+2+4+8+16+32 = 63$, and $2^6-1 = 64-1 = 63$.

3. Grains on the chessboard

10th square: $u_9 = 2^9 = 512$ grains.

Total over the first 10 squares: $S_9 = 2^{10}-1 = 1{,}024-1 = 1{,}023$ grains.

Over 64 squares: $S_{63} = 2^{64}-1 \approx 1.844\times 10^{19}$ grains.

4. Reading and writing large numbers

$65{,}536$ (“sixty-five thousand five hundred thirty-six”)

$S_{63}$ = eighteen quintillion, four hundred forty-six quadrillion, seven hundred forty-four trillion, seventy-three billion, seven hundred nine million, five hundred fifty-one thousand, six hundred fifteen.

5. When growth explodes

When you repeatedly multiply by a number greater than 1, the result grows very fast — exponentially.

Examples:

a bacterial population doubling every hour; the volume in a tank increasing by a fixed percentage; the thickness of a folded sheet doubling each fold…