Problematics | Vizag to Kolkata, via an equilateral triangle

A point lies inside an equilateral triangle. If you know the distance of the point from each of the three vertices, what is the side of the triangle?

Taken together, Google Maps and online calculators are a boon. Thanks to these tools, it was not too difficult to find three locations, each in a different state, that would together form an equilateral triangle.

Welcome to Problematics!(Shutterstock)

Before we proceed to the puzzle, though, here is a word of caution. You may try to work it out using geometry and trigonometry, but I advise you against it. There happens to be a little-known formula that connects the given variables. If you know the formula or can find it (it doesn’t show up easily on Google), the puzzle becomes that much faster.

#Puzzle 40.1

#Puzzle 40.1

The village of Asarar in Satna, MP, is equidistant from the neighbourhoods of Pedamadaka (in Vishakapatnam) and Salt Lake (in Kolkata). Inside this equilateral triangle lies Bhatli (in Bargarh, Odisha), which you can take as being 399 km from Asarar, 455 km from Pedamadaka and 511 km from Salt Lake.

What is the side of the equilateral triangle? Use only the three given distances to calculate the fourth.

#Puzzle 40.2

At the closing stages of the IPL, it took four matches — one qualifier, two eliminators, and the final — to decide a champion among the last four teams. Wimbledon is simpler at three matches (two semis and the final) among the last four.

Take a singles tennis tournament with 100 players. Every match is a knockout, just like Wimbledon. The highest-ranked 28 players are given direct entry to the second round, while the remaining 72 must play in the first round and half of them will advance. 

How many matches must be played across the tournament to decide the winner?

Mailbox: Last Week's Solvers

#Puzzle 39.1

Hi Kabir,

A simple way to look at the problem is to consider everything relative to the water. We can say that relative to the river, the ball is stationary and the boat moves away from the ball and towards the ball at the boat’s independent speed, each leg of the journey taking the same amount of time, that is, 3 minutes either way. In this span of 6 minutes, the ball moves (with the same speed as that of the river) ½km relative to the ground. Thus, the speed of the river is 0.5 km per 6 minutes or 5 km per hour.

The above analysis holds for both days. The boat is rowed for 6 minutes, whether it is rowed downstream first and upstream later, or the other way round. Therefore, the speed of both rivers is 5 km per hour.

— Anshul Kumar, Delhi

[To add to Anshul Kumar’s approach, here’s an analogy. If you drop an object from a moving car, continue to travel for 3 more minutes, and then drive back at the same speed, it will take you 3 more minutes to reach the object. Strictly speaking, the locations of the car and the object are constantly changing due to the Earth’s motion. But because the Earth’s motion has the same effect on both, we can treat the Earth as being stationary when dealing with the car and the object relative to each other. The same principle holds for the boat and the ball relative to each other in a moving river.]

#Puzzle 39.2

Hi Kabir,

The larger pumpkin is a better bargain. It is 20% larger in terms of circumference, thus its radius is 20% larger. Therefore, its volume would be 72.8% more. Thus, the mass i.e. the weight of the larger pumpkin would be 72.8% more while the price is only 50% more.

— Manthan Dhabriya, Mumbai

Solved both puzzles: Anshul Kumar (Delhi), Manthan Dhabriya (Mumbai), Shishir Gupta (Indore), Rajesh Bansal (Noida), Rohit Gurnani (Mumbai), Bharatkumar Chhajer (Mumbai), Shankar Subramanian (Mumbai), Akshay Bakhai (Mumbai), Saurish Seksaria (Mumbai), Amar Lal Mignani (Mumbai), Sunita & Naresh Dhillon (Gurgaon), Gaurav Gummaraju (Navi Mumbai), Amardeep Singh (Meerut), Vikas Nanda

Solved #Puzzle 39.1: Anirudh Maheshwari (Gurgaon)

Solved #Puzzle 39.2: Dr Sunita Gupta (Delhi), Kanwarjit Singh (Delhi), Vivek Arya (Noida), Aarika Goel (Gurgaon), Puneet Vashistha (Delhi), Anand Unnikrishnan (Navi Mumbai), Y K Munjal (Delhi), Geetansha Gera (Faridabad), Anju Rai (Delhi), Sandeep Bhateja (Shimla), Jawahar Lal Aggarwal (Ghaziabad), Anil Khanna, Shreyansh Jain, Pragya Srivastava

Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com