New Delhi: On the occasion of National Mathematics Day on December 22, we shared five interesting puzzles for you to solve. Here are the solutions to the questions.


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?


Answer: The following numbers can be expressed as the sum of two cubes in two different ways: 


4104 = 2³ + 16³ = 9³ + 15³


13832 = 20³ + 183 = 24³ + 2³


20683 = 10³ + 27³ = 19³ + 24³


These are apart from the Hardy-Ramanujan number 1729, which is the smallest such figure also known as Taxicab numbers.


National Mathematics Day Puzzle #2 


Three persons are walking through a desert. X is carrying 15 litres of water, and Y has 9 litres, but Z broke his bottle and is left with no water. So X and Y pool their water, and the three share the 24 litres equally.


When they finally cross the desert, Z pays his companions Rs 800 for sharing their water. But after he leaves, X and Y start fighting. 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.


Answer: Z drank 8 litres of water and paid Rs 800, or Rs 100 per litre. Now let us see how much X and Y actually contributed to Z's share. X contributed 15 litres to the pool but drank 8 litres himself, so only 7 litres from his contribution went to Z. Y contributed 9 litres and drank 8, leaving only 1 litre for Mr Z.


A fair distribution, therefore, would be Rs 700 for X and Rs 100 for Y. When X suggested a division in the ratio 5:3, he was actually asking for less than he deserved, because this would have given him only Rs 500.


National Mathematics Day Puzzle #3 


In this puzzle inspired from a book by maths wizard Shakuntala Devi, we asked an interesting arithmatic question. 


The Delhi-Agra Expressway is 165 kms long. The distance varies on other highway routes but is always more than 200 km. Assuming there is a road of exactly 200 km between Delhi and Agra, you take that route for a drive. You keep a spare tyre with you and decide that even you will use it if you don't have a puncture.


You also decide that you will use the tyres in rotation and change them in such a way that each tyre will have travelled exactly the same distance at the end of the journey.


Question: If you follow this plan, what will be the total distance travelled by each tyre?


Answer: This is a very easy puzzle, but there is a chance of people getting it wrong by dividing the distance of 200 km by the number of tyres, 5, which gives them 40 km.


Remember, the car travels 200 km, but it has 4 tyres at any given time, so the tyres travel a total distance of 800 km. Since this distance is divided equally among 5 tyres, each tyre travels 160 km.


National Mathematics Day Puzzle #4 


Recreational mathematics writer Boris A Kordemsky in his book 'The Moscow Puzzles' shares a puzzle at the end of his narration of a conversation between an idler and the devil.




The devil tells the idler he will double the latter's money every time he crosses a certain bridge. The devil says he will continue to pay every him the idler crosses the bridge but on one condition: the idler must pay him 24 roubles after each crossing.


The idler finds his money doubled after crossing the bridge the first time and gives the devil 24 roubles. He goes for a second spin, and his money is doubled again, and out of that he pays the devil 24 roubles. After the third time, however, though the idler finds his money doubled, he is left with exactly 24 roubles. The devil takes it and leaves.


Question: How much money did the idler have to begin with?


Answer: This can be solved with equations, but it's easier to do it with plain arithmetic, by working backwards.


After the third crossing, the idler had 24 roubles, which means that he had begun with 12 roubles. This was after he had given the devil 24 roubles at the end of the second crossing, which means he had 12 + 24 = 36 roubles before making that payment. So, he had begun the second crossing with half of 36, or 18 roubles.


These 18 roubles were what he had left after paying the devil 24 roubles at the end of the first crossing. So, he had ended the first crossing with 18 + 24 = 42 roubles. In other words, he had begun with 21 roubles.


National Mathematics Day Puzzle #5


Two sharks in the ocean are about to fight after spotting each other from a distance 12 km apart. Let's ignore the fact if it's possible for a shark to see and identify another shark so far away. A sea bird above one of the sharks can also see the other shark. As the first shark races towards the other at a speed of 35 km/hour, the bird too flies towards the second shark.


At a flight speed of 80 km/hr, the bird reaches the second shark much before the first shark.


Also to be kept in mind is the fact that the second shark is also racing towards the first, but at a speed of 25 km/hr, and they began the race at exactly the same moment, as did the bird.


Question: What Is the total distance travelled by the bird?


Answer: Instead of trying to calculate the distance travelled by the bird at each leg of its flight, it is easier to consider the sharks first. When they head towards each other, their combined speed is 35 + 25 = 60 km/hour. Since the distance between them is 12 km, they meet each other after 12/60 = ⅕ of an hour.


The bird travelled for exactly the same amount of time, or ⅕ of an hour. At 80 km/hour, the total distance it flew was 80 x ⅕ = 16 km.