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🌾 Wheat & the Chessboard

The ancient legend that reveals the staggering power of exponential growth

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🌾 The Legend
Topics: geometric sequences, powers of 2, exponential growth, geometric series
A king wanted to reward the inventor of chess. The inventor made a humble request:

"Place 1 grain of wheat on the first square of the chessboard. Place 2 grains on the second square. Place 4 grains on the third — double the previous square each time — until all 64 squares are filled."

The king laughed, thinking the request was trivially small.

How many grains of wheat would actually be needed?
Square n holds 2n−1 grains.
Total = 1 + 2 + 4 + … + 263 = 264 − 1

General formula: Sum of geometric series = a(rⁿ − 1)/(r − 1)
Here: a = 1, r = 2, n = 64 → Total = (2⁶⁴ − 1)/1 = 2⁶⁴ − 1 ≈ 1.845 × 10¹⁹
SquareGrains on this squareRunning total
11 = 2⁰1
22 = 2¹3
34 = 2²7
48 = 2³15
516 = 2⁴31
10512 = 2⁹1,023
32≈ 2.15 × 10⁹≈ 4.29 × 10⁹
64≈ 9.22 × 10¹⁸≈ 1.845 × 10¹⁹
📝 Questions
1
Write an expression for the number of grains on square n (where n = 1, 2, 3 …). Verify it works for n = 1, 2, and 5. 2 marks
2
How many grains are on the 32nd square? Give your answer as a power of 2 and as an approximation. 2 marks
3
Write the total number of grains as a geometric series. Use the formula for the sum of a geometric series to find the exact total. 4 marks
4
The total is approximately 1.845 × 10¹⁹ grains. World wheat production is roughly 7.8 × 10¹¹ grains per year. How many years of global wheat production would be needed to fill the board? 3 marks
5
After which square does the running total first exceed 1 billion (10⁹) grains? Show your working. 3 marks
6
Notice that the number of grains on square n equals (total of all previous squares) + 1. Prove this algebraically. 3 marks
7
Extension: If instead of doubling, the inventor asked for triple the grains each square (1, 3, 9, 27…), write the new total as a geometric series and calculate it. Compare the two totals. 5 marks

Answer Key

🌾 Wheat and Chessboard

1.Grains on square n = 2n−1. n=1: 2⁰=1 ✓. n=2: 2¹=2 ✓. n=5: 2⁴=16 ✓.
2.Square 32: 2³¹ = 2,147,483,648 ≈ 2.15 × 10⁹.
3.Total = Σ 2k−1 (k=1 to 64) = 1+2+4+…+2⁶³. Using S = a(rⁿ−1)/(r−1) with a=1, r=2, n=64: S = (2⁶⁴−1)/(2−1) = 2⁶⁴ − 1 = 18,446,744,073,709,551,615.
4.1.845×10¹⁹ ÷ 7.8×10¹¹ ≈ 23,654 years of global wheat production.
5.Running total after n squares = 2ⁿ − 1. 2ⁿ − 1 > 10⁹ → 2ⁿ > 10⁹ + 1. log₂(10⁹) = 9/log₁₀(2) ≈ 29.9. So n = 30 (total after 30 squares = 2³⁰ − 1 ≈ 1.07 × 10⁹).
6.Sum of first n−1 squares = 2n−1 − 1. Grains on square n = 2n−1. So grains on square n = (sum of previous) + 1. ✓
7.Triple: Total = (3⁶⁴ − 1)/(3−1) = (3⁶⁴ − 1)/2 ≈ 1.69×10³⁰/2 ≈ 8.46×10²⁹. This is roughly 4.6×10¹⁰ times larger than the doubling total.

Answer Key

Wheat and Chessboard

1.2n−1
2.2³¹ ≈ 2.15 × 10⁹
3.2⁶⁴ − 1 ≈ 1.845 × 10¹⁹
4.≈ 23,654 years
5.Square 30 (total = 2³⁰ − 1 > 10⁹)
6.Sum of first n−1 = 2n−1 − 1; grain on square n = 2n−1 = (sum)+1.
7.(3⁶⁴−1)/2 ≈ 8.46×10²⁹.
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