Problematics: The Del-Mu-Ka triangle

This week, Problematics explores a tiny village in Odisha and its unique bond with Delhi and Mumbai.

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Delhi, Mumbai and Kadagabali form an equilateral triangle. What is the side of the triangle? (Source: Express photo)

The village of Kadagabali in Odisha is a dot on the map in almost a literal sense, so tiny that it could provide only 311 names to census officials who came calling in 2011. Its location, nevertheless, makes it momentous in a mathematical sense. Kadagabali, you see, forms an equilateral triangle with Delhi and Mumbai — or so I figure after a search of Google Maps with the help of an online calculator.

Inside this equilateral triangle we need a second key spot at a specified distance from each vertex. By a stroke of coincidence, a point meeting these specifications turns up in a prominent city, Bhopal. For the sake of creating a neat puzzle entirely with integers, however, that I have deliberately introduced small errors (within 5%) to those three distances.

Puzzle#11A: Take Bhopal as 570 km from Delhi, 650 km from Mumbai and 730 km from Kadagabali. Can you now calculate the straight-line distance between Delhi and Mumbai? You may ignore the earth’s curvature.

What you wrote

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Of the six letters below, the writers of the first five have solved all three of last week’s cricket puzzles.

Kabir, this my answer to Puzzle#10A(i). The parity would be restored if the batsmen cross over before the catch is completed. It would not be restored if the batsmen do not cross over.

Sathya Prakash (software developer, New Jersey)

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In Puzzle#10A(ii), the batsmen ran 2 runs but the first run was short, so one run was deducted.

Anirudha Huslurkar (IIT Roorkee)

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Mr Kabir, it took me less than 2-3 minutes to solve the whole set of puzzles. The true story of Puzzle#10A(ii) happened on May 1st in the IPL match between DD and KXIP. The batsmen were Sehwag and Vohra. Though Sehwag ran 2 and came back to the batting end, the second run was not granted as the batsman was short of the crease.

Sampath Kumar V (IIM Kozhikode alumnus)

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In Puzzle#10B, the strike bowler can bowl 10 – x overs more. Since he is supposed to bowl the 50th over, the bowler(s) from the other end will bowl 10 – x – 1 overs. Thus, the captain needs to give the bowling to the strike bowler after 50 – (10 – x) – (10 – x -1) = 31 + 2x overs.

Sanjay Gupta (New Delhi)

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The bowler will be brought back after 31 + 2x overs. The question asks for the over’s number, which is the next one after the above, i.e. 32 + 2x.

Ayanansha Acharya (Kolkata)

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Kabir, I’ve been out of touch with cricket for way too long to understand the loopholes in the game that will help me figure out the solution to the first two problems. For the third one, the answer is 32 + 2x.

Biren Parmar (PhD student, Texas A&M University)

The highway triangle

Puzzle#11B: Two bypasses together encircle Delhi. From point A, two straight highways meet this circle tangentially at P and Q, so that AP = AQ = 100 km. Connecting road BC, a third tangent, meets the circle at R. A motorist takes the northern straight from A to B, uses the connector to reach the southern straight, and drives from C back to A. Find the length of his triangular journey.

Please mail your replies to: kabir.firaque@expressindia.com