Problematics: Railway berth to outer space

This week, Problematics invites readers to a little arithmetic inside the coach, and a little heavenly geometry.

Written by Kabir Firaque | New Delhi | Updated: June 25, 2015 2:30 am
Two heavenly isosceles triangles as seen from North America. (Source: Video grab from NASA website; image edited to the extent of moving the text within each frame)

Ten out of fifteen @ 2:3. That’s how frequently Problematics has so far related its puzzles to current events, sometimes naturally, sometimes loosely. In keeping with that unnecessary but healthy trend, both puzzles this week have some kind of topical connection.

The first one is about railway berths. Ladders are coming up inside coaches to make it easier to climb into the upper bunks, according to this report by Avishek G Dastidar. Designers may be starting with AC-I coaches, which are used by VVIPs and other assorted bores, but will go on one day into the aam aadmi’s three-tier coach, which is oriented like this:

Problematics 2

A confirmed ticket specifies whether a berth is lower, middle or upper. For travellers who begin on the waiting list, the best place to check is the railway website. If not, wait until the last hour when the list of newly confirmed berths is pasted on the coach.

Jostling with the crowd, you find berth 37 against your name. “Middle,” you announce, with a quick mental calculation. “Mine’s 46,” says another passenger, and you help him: “That’s upper for you.” And when a third passenger asks, “Where is my 55?” you reply at once: “Side lower.” As a crowd of 71 gathers in admiration, you choose not to tell them how easily it is done.

Puzzle#16A: How is it done? This question is not addressed to people who post answers on Facebook, only to people who write emails to Problematics. But over to the next puzzle.

Problematics 3

Jupiter and Venus have been flirting scandalously this month, with the crescent Moon completing a love triangle. The above orientation, which I have traced from a NASA video and scaled up manually, is from a North American perspective. On both days, Jupiter, Venus and the crescent Moon formed a perfect isosceles triangle. Those heavenly occasions, however, are only an excuse to talk about isosceles triangles in general.

Back in school, they taught us that if side AB = side AC, then angle A = angle B. To prove this natural property of isosceles triangles, they made us drop a line from A to D, the midpoint of BC, prove that the smaller triangles on either side of AD are congruent, hence base angles equal.

There is, however, a much prettier proof.

Puzzle#16B: No additional construction such as line AD. No trigonometry. Just you, geometry and isosceles triangle ABC. Prove that the base angles are equal.

What you wrote
Your solutions to last week’s puzzles, starting with the second question first.

Problematics 4

Mr Kabir, for Puzzle#15B, 8th day is the answer when the number of reads on Facebook crosses 1.44 billion.
— Sampath Kumar V (IIM Kozhikode alumnus)
***
Hi Kabir, for Puzzle#15A, after writing too many equations and buzzing my head out, I googled “perfect squares in Fibonacci sequence” and found a bigggg proof of how there are no other perfect squares in a Fibonacci sequence other than 1, 144. That proof is too complex.
— Bindia George (Kochi)

Too complex indeed, Bindia. For this blog, just the statement is enough — no perfect square in Fibonacci sequence higher than 144.

Solved both puzzles: Sanjay Gupta (New Delhi), Biren Parmar (Texas A&M University), Vivek Jalan (founder, Customate Systems), Bindia George, Sampath Kumar V

Answered Fibonacci puzzle: Sathya Prakash (New Jersey), Anil Kumar

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

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