Problematics | Watch your hands and how they move

Jan 10, 2023 03:34 PM IST

Ignore the second hand and concern yourselves only with the hour and minute hands. Like last week’s Christmas tree puzzle, this one too comes from something I had presented to readers 30 years ago.

Timekeeping may be largely digital these days, but the old-fashioned watch (and clock) remains in use, if sometimes merely for fashion. There are puzzles to be found in digital displays too, and I am indeed working on creating one around the faulty display outside the lift in my apartment, but watches with separate hands for hours, minutes and seconds offer a wider range of possibilities.

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

Ignore the second hand and concern yourselves only with the hour and minute hands. Like last week’s Christmas tree puzzle, this one too comes from something I had presented to readers 30 years ago.

Problematics Week 20.
Problematics Week 20.
#Puzzle 20.1:
You know that when the day begins at midnight, the hour and minute hands will coincide at 12 on the dial of your watch. This will happen again at noon. In between, the minute hand will catch up with and overtake the hour hand several more times, and there will be moments when the two hands are at exactly the same place.
As a warm-up, list the exact times between midnight and noon when the two hands coincide. Once done, enjoy the main puzzle below.
In a faulty watch, the hour hand works fine, but the minute hand moves anticlockwise at a constant speed. The two hands meet every 80 minutes. If the last time the watch gave you the correct time was at 12 noon, when will the next such time be?
#Puzzle 20.2.
#Puzzle 20.2.
Mailbox: Last week’s solvers:

#Puzzle 19.1:

Hello Sir,  

Let us consider the time taken by all three combined to be 'x'.  

This means the Andhra man takes (x + 1) hours, the Bihar man takes (x + 6) hours and the Chhattisgarh man takes (2x) hours.

1/(x + 1) + 1/(x + 6) + 1/(2x) = 1/x

=> (2x + 7)/(x² + 7x + 6) = 1/2x

=> 4x² + 14x = x² + 7x + 6  

=> 3x² + 7x – 6 = 0

Solving the quadratic equation,

x = 2/3 hours or 40 minutes.

Therefore, time taken by the Andhra man = 1 hr 40 min; time taken by Bihar man = 6 hr 40 min; and time taken by Chhattisgarh man = 1 hr 20 min.

Hence, you chose the Chhattisgarh person who did the work in 1 hr 20 min.

Mayobhav Pathak, Gurgaon

#Puzzle 19.2:

Hi Kabir,

1kg, 3kg, 9kg, 27kg

Start with 1kg and 3kg. We can weigh between 1kg and 4kg using these 2 weights. For 5kg, let's assume all of the smaller weights are together in the pan along with the 5kg of material being weighed. Total weight needed to balance this on the other side will be 5 + 3 + 1 = 9kg. We can verify that using these 3 weights, we can measure all weights between 1kg and 13kg. Using similar logic for 14kg, we need 14 + 9 + 3 + 1 = 27kg. Again, it can be verified that all weights between 1kg and 40kg can be weighed using this combination.

Biren Parmar, Bay Area, California

(Yojit Manral found this puzzle too easy and has offered a similar one, which I shall present to readers some other time.)
Solved both puzzles: Mayobhhav Pathak (Gurgaon), Avneesh Kumar (Delhi), Gulraj Nagi (Ludhiana), Yojit Manral (Mumbai), Biren Parmar (Bay Area, California), Mahesh Mundhra (Indirapuram), Sandeep Bhateja (Hoshiarpur), Prakash R Gangaramani, Naresh Dhillon (Gurgaon), Dr Gurbachan Lal Arora (Delhi), Dhairyan Jain (Mumbai), Lucky Singh Randhawa (Delhi), Anil Kumar Goyal (Delhi), Gopal Menon (Mumbai), Tanmay Vakharia, Rahul Agarwal (Bay Area, California), Dr Roona Poddar (Delhi University), Amardeep Singh (Meerut), Ravi Sondhi & Rudra Sondhi (Gurgaon), Jaikumar Bhatia (Ulhasnagar, Thane), Shishir Gupta (Indore), Jasvinder Singh (Nabha)
#Solved Puzzle 19.1: Annu Singhal, Yuganshi Jain (Delhi), Madhuri Patwardhan (Thane), Avanti Kashikar
#Solved Puzzle 19.2: Naresh Kumar Mendiratta, Abhishek Garg (Chandigarh), Aarika Goel (Gurgaon), Sumit Sahni (Delhi), Prakash R Gangaramani, Sandeep Ohlan (Rohtak), Ira Srivastava , Rina Kumari (Noida Extension)

Problematics will be back next week. Please send in your replies 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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