Problematics | Random walks with a point in common - Hindustan Times

Problematics | Random walks with a point in common

Sep 25, 2023 05:07 PM IST

A puzzle seemingly impossible to solve, but in fact deceptively simple. Can you solve it without using mathematics?

My old friend Arijit Das Gupta gave me an unusual puzzle 30 years ago, when we were colleagues. I have never seen it published in any form except on the two occasions when I ran it myself, the first time in a daily in my hometown in 1993, and then online in a multi-edition newspaper in 2015.

Welcome to Problematics!(Shutterstock) PREMIUM
Welcome to Problematics!(Shutterstock)

The puzzle looks impossible to solve but is deceptively simple. You cannot solve it with mathematics; logic is all you need.

#Puzzle 57.1
#Puzzle 57.1

The following morning, he starts downhill at 7am. Although it’s an easier trip, he takes his time, taking in the scenery and resting his tired muscles. The durations of resting and walking, and his inconsistent speeds along the descent, are once again random. One thing is common, though: he follows the exact same path both ways, every point of the journey. He reaches the bottom at 7pm.

Prove that there is a point on the route where he was present at the same time on both days.

#Puzzle 57.2
#Puzzle 57.2

#Puzzle 57.2

This puzzle, too, is a reminder from the good old days: it comes from a book I had bought in the 1980s. The book includes the solution but not a step-by-step process for arriving at it, so do describe your methods when you send in your answers.

You have a ribbon measuring 12.5 feet, with which you wish to tie a gift box. You want the ribbon to go around the box’s length once and around its width twice, as shown in the illustration. The knot will take up to 6 inches.

What is the largest box in terms of volume that you can tie this way?

Mailbox: Last week’s solvers

#Puzzle 56.1

Hi Kabir,

Time taken to text an invitation was 10 minutes and to text a cancellation was 20 minutes. Time taken to email an invitation was 15 minutes and to email a cancellation was also 15 minutes. Together, invitation and cancellation times add up to 30 minutes per guest, whether text or email. Since sending all the invitations and cancellations took 12 hours (720 minutes), the number of guests is 720/30 = 24.

The time spent in sending the invitations was a certain number of hours minus 5 minutes, or 60h – 5 minutes, where h is the number of hours. This value has to be between 240 minutes (all invitations by text) and 360 minutes (all invitations by email). Therefore, h can be either 5 or 6.

Solution 1 (h = 5): 13 text invitations and 11 email invitations.

Solution 2 (h = 6): 1 text invitation and 23 email invitations.

— Professor Anshul Kumar, Delhi


Hi Kabir,

#Puzzle 56.1 talks of “Relatives and friends responded to the cancellation texts” — everywhere plural. So, only 1 text is not possible. I would go with 13 texts and 11 mails, even though 1 text and 23 mails are mathematically possible.

— Akshay Bakhai, Mumbai

#Puzzle 56.2

Hi Kabir,

There are many possible answers, but I wanted to use prime numbers as far as possible:

(17/89) + (504/623) = 1

— Dr Nakul Makkar, Noida

Indeed, there are many possible answers, and it’s impossible to list them all. Anyone who has sent at least one pair of fractions is acknowledged below.

Solved both puzzles: Prof Anshul Kumar (Delhi), Akshay Bakhai (Mumbai), Dr Nakul Makkar (Noida), Dr Sunita Gupta (Delhi), Anil Khanna (Ghaziabad), Arun Kumar Gupta (Greater Noida), Sunita & Naresh Dhillon (Gurgaon), Group Capt RK Shrivastava (retd; Delhi), Amardeep Singh (Meerut), V Anand (Noida), Sankaran KB (Chennai), Shri Ram Aggarwal (Delhi), Rachna Jain (Delhi), Vikas Nanda (Pathankot), Richard D’Souza (Mumbai), Harsh Ozare (Kalyan)

Solved #Puzzle 56.1: Kanwarjit Singh, Harshit Arora, Shawn Jacob, SC Vasudeva

Solved #Puzzle 56.2: Yamini, Bhasker Mundhra, Yadvendra Somra, Charvi Brajpuriya, Aishani Kothiwal, Sushma Pandhi, Bhuvi Jain, Ajay Ashok, Nipun Bamania, Sabornee Jana, Soumil Mukhopadhyay, Nirmit Kohli, Jaikumar Bhatia & Disha Bhatia

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

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    Puzzles Editor Kabir Firaque is the author of the weekly column Problematics. A journalist for three decades, he also writes about science and mathematics.

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