Mystery of Missing Money – 1 Rupee by Shakuntala Devi
Two women sell marbles at a market. One charges 3 marbles for ₹1. The other charges 2 marbles for ₹1. They each have 30 unsold marbles at the end of the day.
Instead of selling separately, they combine their marbles and ask a friend to sell all 60 at “5 for ₹2.” Seems fair, right? 3 + 2 = 5. And ₹1 + ₹1 = ₹2.
But when they count the money, they’re ₹1 short. The puzzle: where did that rupee go?
This classic problem comes from Shakuntala Devi, India’s “Human Computer.” It’s a perfect example of how intuitive math can trick us.
The Puzzle
Two women sell marbles in the marketplace. The first sells 3 marbles for ₹1. The second sells 2 marbles for ₹1.
One day, each has 30 unsold marbles. Rather than pack up, they decide to combine their stock and give all 60 marbles to a friend. She’ll sell them at 5 marbles for ₹2.
Their logic seems solid. If one sells at 3 for ₹1 and the other at 2 for ₹1, then selling 5 for ₹2 should be equivalent.
Here’s what they expected to earn if they’d sold separately:
First woman: 30 marbles at 3 for ₹1 = ₹10
Second woman: 30 marbles at 2 for ₹1 = ₹15
Total expected: ₹25
But when the friend sells all 60 marbles at 5 for ₹2, she collects only ₹24.
That’s \( 60 \times \frac{2}{5} = 24 \) rupees.
One rupee vanished. Where did it go?
The Solution
There’s no mystery. The math just doesn’t work the way our intuition suggests.
The two selling rates (3 for ₹1 and 2 for ₹1) can only be combined into “5 for ₹2” if the marbles are mixed in a 3:2 ratio. That’s 36 marbles from the first woman and 24 from the second.
But they contributed equal amounts. 30 each. That breaks the required ratio.
Why the Ratio Matters
Think about what each marble is actually worth.
First woman’s marbles: ₹1 ÷ 3 = ₹0.333 per marble
Second woman’s marbles: ₹1 ÷ 2 = ₹0.50 per marble
Combined rate: ₹2 ÷ 5 = ₹0.40 per marble
Here’s the problem. The combined rate of ₹0.40 isn’t the average of ₹0.333 and ₹0.50. The true average would be ₹0.4167.
By selling at ₹0.40 per marble, they’re underpricing the second woman’s marbles significantly. She loses ₹0.10 on each of her 30 marbles. That’s ₹3 lost.
Meanwhile, the first woman gains ₹0.067 on each marble. That’s ₹2 gained on her 30 marbles.
Net result: ₹3 lost minus ₹2 gained = ₹1 missing.
The Pattern
This ₹1 loss happens for every 60 marbles sold when contributing equal amounts.
If they had 60 marbles each (120 total), they’d lose ₹2. With 90 each (180 total), they’d lose ₹3. And so on.
The only way to avoid any loss is to maintain that 3:2 contribution ratio. If the first woman gives 36 and the second gives 24, the combined rate works perfectly. Both would earn exactly what they would have earned selling separately.
The Lesson
You can’t just add fractions by adding numerators and denominators.
\( \frac{1}{3} + \frac{1}{2} \neq \frac{2}{5} \)
The correct sum is \( \frac{1}{3} + \frac{1}{2} = \frac{5}{6} \), which means 5 marbles should cost ₹\( \frac{6}{5} \) or ₹1.20 each pair, not ₹2 for 5.
This puzzle trips people up because “3 for 1, plus 2 for 1, equals 5 for 2” sounds right. But it only works when quantities match the price ratio.
About Shakuntala Devi
Shakuntala Devi earned the nickname “Human Computer” for good reason. She could multiply two 13-digit numbers in her head in under 30 seconds. No calculator. No paper.
Born November 4, 1929, in Bangalore, she showed extraordinary arithmetic abilities as a child. By age 6, she was demonstrating mental calculations at university events.
Her 1980 multiplication feat (multiplying 7,686,369,774,870 by 2,465,099,745,779 in 28 seconds) landed her in the Guinness Book of World Records. The answer, by the way, was 18,947,668,177,995,426,462,773,730.
Beyond mental math, she wrote books on puzzles, arithmetic shortcuts, and astrology. The marble puzzle above comes from her collection of mathematical brain teasers.
She passed away in 2013, but her puzzles keep teaching people that math intuition can fool you.
Frequently Asked Questions
Where did the missing rupee actually go?
The rupee didn’t go anywhere. It was never there. The assumption that “3 for ₹1 plus 2 for ₹1 equals 5 for ₹2” is mathematically wrong when equal quantities are combined. The second woman’s marbles were underpriced by ₹3 while the first woman’s marbles were overpriced by ₹2, creating a net ₹1 shortfall.
Who was Shakuntala Devi?
Shakuntala Devi (1929-2013) was an Indian mathematician known as the “Human Computer.” She could perform complex mental calculations faster than early electronic computers. She held a Guinness World Record for multiplying two 13-digit numbers in 28 seconds and authored several books on mathematics and puzzles.
Why doesn’t 3 for ₹1 plus 2 for ₹1 equal 5 for ₹2?
Because you can’t add fractions by adding their numerators and denominators. The price per marble is ₹0.333 and ₹0.50 respectively. The average of these is ₹0.4167, not ₹0.40 (which is what 5 for ₹2 gives you). This pricing error causes the ₹1 loss.
What ratio of marbles would avoid the loss?
A 3:2 ratio. If the first woman (selling 3 for ₹1) contributes 36 marbles and the second woman (selling 2 for ₹1) contributes 24 marbles, the combined “5 for ₹2” rate works perfectly. Both women would earn exactly what they’d have earned selling separately.
How much does each woman lose in this puzzle?
If they split the ₹24 proportionally to what they expected (₹10 and ₹15), the first woman gets ₹9.60 (loses ₹0.40) and the second gets ₹14.40 (loses ₹0.60). If split equally at ₹12 each, the first woman gains ₹2 while the second loses ₹3.
Is this similar to other missing money puzzles?
Yes. It’s related to puzzles like the “missing dollar” hotel problem. These puzzles exploit our tendency to add numbers incorrectly or track money flows in misleading ways. The “missing” amount was never actually there to begin with.
What mathematical concept does this puzzle teach?
Weighted averages. You can’t simply average two rates without considering the quantities involved. When combining rates, you must weight each rate by its corresponding quantity. This applies to everything from pricing to calculating average speeds.
Would the loss increase with more marbles?
Yes. The loss is ₹1 for every 60 marbles when contributed equally. With 60 marbles each (120 total), they’d lose ₹2. With 90 each (180 total), they’d lose ₹3. The loss scales proportionally with quantity.
What was Shakuntala Devi’s Guinness World Record?
In 1980, she multiplied 7,686,369,774,870 by 2,465,099,745,779 in just 28 seconds. The answer was 18,947,668,177,995,426,462,773,730. This feat earned her a place in the 1982 Guinness Book of World Records as the fastest human computer.
Where can I find more puzzles like this?
Shakuntala Devi wrote several puzzle books including “Puzzles to Puzzle You,” “More Puzzles to Puzzle You,” and “The Book of Numbers.” These contain similar brain teasers that challenge mathematical intuition and reveal common reasoning errors.