Problem #97 EASY
The Wheat and the Chessboard
Paradox Exponential Growth Sequences Math
Problem Statement
According to legend, the inventor of chess presented his game to a king who was so delighted he offered any reward. The inventor asked for one grain of wheat on the first square of the chessboard, two on the second, four on the third, doubling each time for all 64 squares. The king laughed and agreed, thinking it a modest request. How many grains of wheat does the inventor receive in total? And how does this compare to all the wheat ever produced in human history?
Answer & Quick Explanation
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Total = 2⁶⁴ − 1 ≈ 18.4 quintillion grains ≈ 922 billion tonnes. At current global wheat production (~775 million tonnes/year), repaying the debt would take approximately 1.19 million years. The king was ruined by the mathematics of doubling.
WOW Moment:
- Square 1: 1 grain of wheat.
- Square 32: ~100 tonnes. A full truckload.
- Square 48: ~140 million tonnes. More than entire India yearly harvest.
- Square 64: 9,223,372,036,854,775,808 grains on the LAST square alone.
- Total: ~922 billion tonnes of wheat.
- Earth's total wheat production since farming began ~10,000 BC is roughly 4.5 trillion tonnes.
- The king owes about 200 years of ALL food ever grown on Earth.
- The king thought he was giving away a bag of rice. He had actually promised more food than humanity has produced in all of recorded history. Doubling 64 times is not 'a lot.' It is incomprehensible.
Detailed Editorial Solution
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The story of the wheat and the chessboard is the classic illustration of the power of exponential growth. Human intuition is linear; we struggle to comprehend how quickly numbers grow when they are repeatedly multiplied.
Let's look at the mathematical formulation:
1. The number of grains on each square forms a geometric progression:
- Square 1: 2^0 = 1
- Square 2: 2^1 = 2
- Square 3: 2^2 = 4
- ...
- Square 64: 2^63
2. The total number of grains is the sum of this geometric series:
S = 2^0 + 2^1 + 2^2 + ... + 2^63
3. Using the formula for the sum of a geometric series:
S = (a * (r^n - 1)) / (r - 1)
where a = 1 (first term), r = 2 (common ratio), and n = 64 (number of terms).
S = (1 * (2^64 - 1)) / (2 - 1) = 2^64 - 1.
4. Calculating the exact value of 2^64 - 1:
S = 18,446,744,073,709,551,615 grains.
This is approximately 18.4 quintillion grains.
To explain the WOW part:
- A single grain of wheat weighs about 50 milligrams (0.05 grams).
- The total weight of 18.446 × 10^18 grains is:
Weight = 1.8446 × 10^19 × 0.05 grams = 9.223 × 10^17 grams = 922,337,203,685 tonnes.
- To put 922 billion tonnes into perspective, the global annual production of wheat is about 775 million tonnes.
Years of harvest = 922,337 million tonnes / 775 million tonnes ≈ 1,190,112 years.
The king would need the entire agricultural output of the planet for over 1.19 million years to fulfill his promise!
- The most shocking property of this geometric progression is that the last term (2^63 ≈ 9.22 quintillion) is greater than the sum of all preceding 63 terms combined (2^63 - 1). The 64th square alone contains 50% of the entire reward.