Problematics | Watch those hands - Hindustan Times
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Problematics | Watch those hands

Apr 22, 2024 06:43 PM IST

Here’s an old-school puzzle involving the hour and minute hands of a watch. How frequently do they overlap or stretch out diametrically in a straight line?

Nine to five is a sweet phrase and a great concept, but its accuracy is something I have often wondered about. In how many offices does work begin at 9am? And even if it does and everyone arrives on the dot, how often do people leave at exactly 5pm without completing whatever project, assignment or paperwork they are involved in at that moment?

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

That said, we will visit an office where the hours are 9 to 5, but only on paper, of course. The staff do arrive at 9am or thereabouts but they usually get to leave when it’s well past 5pm, sometimes soon after, but mostly long after.

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#Puzzle 87.1

So we have this couple of fitness freaks who work in the same office. Their home is miles away but they prefer to walk rather than drive. .

One evening as they set off for home after 5pm, they observe with interest that the hour and minute hands on their watches are overlapping. They walk at their usual pace and are just about to reach home when the woman notices a look of concern on her husband’s face.

“What’s the matter?”

“I think I left the key in the office,” he replies, desperately searching his pockets once more. “I will have to walk back and fetch it,” he continues, still walking in the direction of their home.

“What do you mean?” says his wife. “If you think you are getting that extra exercise without me, think again.”

They reach their doorstep and turn back without wasting a second.

“We have walked for over an hour and a half,” says the husband, “so it will take us over three hours more to reach office and come back.”

“Have you noticed,” says the wife, “that the minute and hands were exactly opposite each other at the precise moment we reached the door? The two hands stretched diametrically across the dial.”

“Indeed I noticed,” nods the husband. “And if we maintain the same speed, the hands will be overlapping again when we reach the office.”

At what times did the couple leave office and reach home the first time?

#Puzzle 87.2

The couple have a daughter who was born on her mother’s birthday. Today, the mother is twice as old as the daughter; 20 years ago, she was six times as old as the girl was then. It’s easy to determine their ages, but I am wondering if it can be done without using algebra.

If you can, send me an arithmetical solution; if not, use algebra as the last resort.

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 86.1

Dear Kabir,

Let the final score of the No. 5 batter be n and that of No. 8 be m.

Then n = n/7 + n/5 + 5 + 2m + 4, which gives 23n/35 = 2m + 9

n is a multiple of 7*5=35. Let n be 35x.

So, 23n/35 = 2m + 9 gives 23x = 2m + 9; or m = (23x – 9)/2

Now m < 50, so, 23x < 109, or x < 5

x = 2 and x = 4 give non-integer values for m. Therefore x can be either 1 or 3.

x = 1 gives m = 30 and n = 105

x = 3 gives m = 7 and m = 35.

— Yadvendra Somra, Sonipat

#Puzzle 86.2

For the word puzzle, which as I mentioned comes from a book, I was always doubtful about the sweeping claim that just four English words end in the letters “DOUS”. I never bothered to check, though, until I finally turned it over to you readers last week. Not surprisingly, many of you came up with long lists of words, many of them biological or related to chemical elements.

Dear Kabir,

I think you are referring to these 4 words: TREMENDOUS, HORRENDOUS, STUPENDOUS, HAZARDOUS. But I found 47 words from an online search: AMADOUS, AMBIPODOUS, APODOUS, BIOHAZARDOUS, CEPHALOPODOUS, CHILOPODOUS, CYNOPODOUS, DECAPODOUS, FRONDOUS, GASTEROPODOUS, GASTROPODOUS, HAZARDOUS, HORRENDOUS, HYBRIDOUS, IODOUS, ISOPODOUS, JEOPARDOUS, LIGNIPERDOUS, MOLYBDOUS, MULTIFIDOUS, MYRIAPODOUS, NEFANDOUS, NODOUS, NONHAZARDOUS, OCTOPODOUS, OSTRACODOUS, PALLADOUS, PALUDOUS, POLYPODOUS, PUDENDOUS, RHIZOPODOUS, RHODOUS, SAUROPODOUS, SCHIZOPODOUS, SOLIPEDOUS, SPLENDIDOUS, STEGANOPODOUS, STUPENDOUS, TETRAPODOUS, TREMENDOUS, ULTRAHAZARDOUS, UNHAZARDOUS, UNTREMENDOUS, UROPODOUS, VANADOUS, VODOUS, VOUDOUS.

— Sundarraj C, Bengaluru

Solved both puzzles: Yadvendra Somra (Sonipat), Sundarraj C (Bengaluru), Dr Sunita Gupta (Delhi), Anil Khanna (Ghaziabad), Professor Anshul Kumar (Delhi), Kanwarjit Singh (Chief Commissioner of Income Tax, retired), Ajay Ashok (Mumbai), Akshay Bakhai (Mumbai), Y K Munjal (Delhi), Sanjay S (Coimbatore)

Jaikumar Inder Bhatia and Disha Bhatia sent a list of words but did not solve the cricket puzzle. Shishir Gupta said he found 83 words on a website, but did not send a list.

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

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  • ABOUT THE AUTHOR
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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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