Problematics | Lies and truths for the New Year - Hindustan Times

Problematics | Lies and truths for the New Year

Jan 01, 2024 08:27 PM IST

We kick off 2024 with some people who always tell the truth and some others who always lie. Can you determine how many there are of each kind?

The last time New Year's Day fell on a Monday was in 2018 when I wasn't writing Problematics for this paper (I was doing so in 2002). The next time January 1 is a Monday will be in 2029, a slightly smaller gap because there are two leap years in between. Even so, that's five years ahead, so we all know what a rare occasion this is.

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

A very Happy New Year to all my readers. Let's ring it in with a couple of puzzles.

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#Puzzle 71.1


In all the previous 70 weeks, we have had only one puzzle involving people who either always tell the truth or always lie. Here is another of them, one that is more unusual than the previous puzzle such as this.

So, we get introduced to not two such people but a large group of them. As usual, those who speak the truth are always truthful, and liars will always be liars, as everyone knows.

We try to figure out who is what. Our initial questions like "Do you speak the truth?" don't work, because anyone who is asked that always replies yes.

We try another trick. We ask the entire lot to stand in a circle and ask each one: "Is the person on your left a seeker of eternal truth or a liar?" To our unhappy surprise, we find each one replying, "Oh, he/she is a liar."

This was probably not coincidental; they must have stood in that arrangement deliberately to prevent us from figuring out their inherent habits.

Giving up, we head home. On the way, you wonder, "How many of them were there?"

"How careless of us," I reply; "we didn't even count."

"Let's call the guy in the red shirt," you say, whipping out your mobile and putting it on the speaker. "Hi, how many of you were in the group?" you ask. "We were 51," comes the reply.

"That's no good," we muse after you have hung up. "How do we know if he was telling the truth?"

So we call another one of them, the woman in a blue dress. "Hi, how many of you were in the group?" we repeat. "We just called the fellow in the red shirt and he said there were 51 of you."

"You should never have called Redshirt," she replies, "for he's a born liar. There were actually 54 of us."

After the call ends, we note that either one of the two could have been lying. Or maybe both were liars. Then the truth dawns on us.

How many people were in the group?


#Puzzle 71.2


This puzzle is very old and I have come across it in many parts of the country as well as in books published in the UK and the US. It's quite likely that you too have solved it at some point in the past. But this is a celebratory occasion that calls for some light-hearted fun, especially if you have not come across the puzzle before.

Three people share the first floor of a house, paying 30,000 a month as rent. On the New Year, their landlord on the ground floor, in a rare show of generosity, waives 5,000. "Just this once," he tells himself. He then hands ten crisp notes of 500, freshly withdrawn from the ATM, to his young son and asks him to return it to the three tenants upstairs.

On the way upstairs, the boy does a quick calculation: "They'll never be able to share the refund equally. They'll always have to deal with a recurring decimal."

He decides to solve their problem by pocketing four of the notes for himself.

He knocks on the tenant's door. "A refund of 3,000 for you," he hands over six notes and flees lest the tenants start asking too many questions.

So the three friends share 1,000 each. Subtracted from the 10,000 paid earlier, that's a net payment of 9,000 each. A total of 27,000. The landlord's son has 2,000. You don't need a calculator to check that 27,000 + 2,000 = 29,000.

From the original payment of 30,000, where did the remaining 1,000 go?




#Puzzle 70.1

Solution 70.1
Solution 70.1

Hi Kabir,

The first customer had bought exactly twice as much liquor as the second customer bought after him. Possible solutions are shown in the table.

— Professor Anshul Kumar (Delhi)

#Puzzle 70.2

Dear Kabir,

Two possible solutions satisfy the given conditions, as shown in the table.

Solution 70.2
Solution 70.2

The letters stand for:

Option 1: A = 4, E = 7, L = 0, M = 2, O = 1, R = 5, S = 9, T = 8, X = 3, Y = 6

Option 2: A = 9, E = 4, L = 0, M= 3, O = 1, R = 2, S = 6, S = 8, X = 7, Y = 5

Happy New Year.

— Sundarraj C, Bengaluru

Solved both puzzles: Professor Anshul Kumar (Delhi), Sundarraj C (Bengaluru), Neel Bainsla (Delhi), Dr Sunita Gupta (Delhi), Abhishek Garg (Chandigarh), Group Captain RK Shrivastava (retd; Delhi), Shri Ram Aggarwal (Delhi), Ajay Ashok (Mumbai), Shishir Gupta (Indore), Yadvendra Somra (Sonipat), Anil Khanna (Ghaziabad)

Solved #Puzzle 70.1: Radhika Joshi (DPS Vasant Kunj), Jaikumar Inder Bhatia and Disha Bhatia (Ulhasnagar, Thane),

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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