New Delhi: December 22 is National Mathematics Day in India. It is observed to commemorate the birth anniversary of the Indian mathematical genius, Srinivasa Ramanujan. On this occasion, here are 5 mathematical puzzles you can try solving. These are not at all difficult, so most readers should be able to solve them.
Our first puzzle will be related indirectly to what we know as the Hardy-Ramanujan number. The story of the conversation between Ramanujan and the British mathematician GK Hardy is well known. An unwell Ramanujan was in hospital, and Hardy, Ramanujan’s mentor, was visiting him. There, Hardy told Ramanujan that he had come to visit the latter in a taxi with a registration number, ‘1729’, and described it as “rather a dull one”. In response, Ramanujan is quoted to have said: “No Hardy, it’s a very interesting number! It’s the smallest number expressible as the sum of two cubes in two different ways.”
The anecdote led to the discovery of a special number — 1729 — which went on it be called as the Hardy-Ramanujan number. This remains Ramanujan’s most popular discovery to date.
National Mathematics Day Puzzle #1
1729 = 10³ + 9³ = 12³ + 1³.
1729 is the sum of 1000 and 729. 1000 is the cube of 10 and 729 is the cube of 9. Therefore, 1729 is the sum of the cubes 10 and 9.
Also, 1729 can be represented as a sum of 1728 and 1. 1728 is the cube of 12, and 1 is the cube of 1. Therefore, 1729 is also a sum of the cubes of 12 and 1.
Question: Can you find at least three other numbers that can be expressed as the sum of two cubes in two different ways?
National Mathematics Day Puzzle #2
Three persons are walking through a desert. X is carrying 15 liters of water, and Y is carrying 9 liters, but Z has broken his bottle and is left with no water. So X and Y pool their water, and the three share the 24 liters equally.
When they finally cross the desert and reach a town, Z pays his companions Rs 800 for sharing their water with him. But after he leaves, X and Y start quarrelling. X says he should get more because he contributed more water to the pool. “Let us divide it in a 15:9 ratio, or 5:3,” he says. But Y says they should share the money equally because everyone drank the same amount of water.
Question: Calculate the amount that X and Y should get on the basis of the quantity of water they pooled.
National Mathematics Day Puzzle #3
This is a puzzle inspired from a book by another Indian maths wizard, Shakuntala Devi. Here is how it goes.
The length of the Delhi-Agra Expressway is 165 km. If we use other highway routes, the distance between the two cities varies, but is always more than 200 km. Let us assume that there is a road of exactly 200 km between the two cities. One morning, you set off on a drive from Delhi to Agra. After checking the 4 tires of your car, you also keep a spare one. You decide that even if you don’t have a puncture, you will still use the spare.
In fact, being a mathematician, you decide that you will rotate and change the tires in such a way that at the end of the journey, each tire will have traveled exactly the same distance.
Question: If you follow this plan, what will be the total distance traveled by each tire?
National Mathematics Day Puzzle #4
In the book, ‘The Moscow Puzzles’, the recreational mathematics writer Boris A Kordemsky describes a conversation between an idler and the devil, and ends it with a puzzle.
In short, the devil strikes a deal with the idler. If the idler crosses a bridge, the devil will double the idler’s money. If the idler crosses the bridge again, the devil will double his money again. This will go on as long as the idler fulfills one condition: after each crossing, he must pay the devil 24 roubles.
The idler crosses the bridge, finds his money doubled, and gives the devil 24 roubles. After he crosses it a second time, his money is again doubled, out of which he pays the devil 24 roubles. After the third crossing, although his money has been doubled for the third time, the idler finds that he has exactly 24 roubles. He gives it to the devil, who laughs and goes away.
Question: How much money did the idler have to begin with?
National Mathematics Day Puzzle #5
In the wide ocean, two sharks are about to fight after spotting each other from a distance 12 km apart. For the sake of this puzzle, we can ignore the fact whether it is possible for a shark to see and identify another shark so far away. Above one of the sharks is a sea bird, which can also see the other shark. As the shark directly below the bird races towards the other shark at a speed of 35 km/hour, the bird too flies towards the second shark, in amusement.
The bird’s flight speed is 80 km/hr, which means it reaches the second shark much earlier than the first shark arrives. Besides, the second shark too is racing towards the first shark, at a speed of 25 km/hr. In fact, the two sharks began racing towards each other at exactly the same moment, and the bird too started its flight at that very moment.
When the bird reaches the second shark, it wastes no time, turns back in the same instant, and starts flying towards the first shark, maintaining its flight speed of 80 km/hr. When it reaches the first shark, it turns around once again and heads towards the second shark. It keeps repeating this back and forth flight, never wasting a second, until the two sharks finally meet and begin their fight.
Question: What is the total distance traveled by the bird?
Did you find the quiz interesting? Just take out your pen, notebook and calculator, or do the maths in your mind, and write to us if you think you have the right answers. Use hashtag #ABPNationalMathematicsDayQuiz and post your answers on Twitter, Facebook and Koo latest by December 22 midnight. Don’t forget to tag us.
The solutions will be posted on ABP Live on Thursday, December 23.
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