Problematics | Lewis Carroll transported to Haryana - Hindustan Times

Problematics | Lewis Carroll transported to Haryana

Jan 29, 2024 08:00 AM IST

The creator of Alice in Wonderland was also a mathematician and master puzzler. To mark his birth anniversary, here is an adaptation of one of his puzzles

The 192nd birth anniversary of one of the world’s pioneering puzzlers passed during the weekend. Born on 27 January 1832, he will always be known as Lewis Carroll, which was actually a pen name (as mentioned in these columns earlier). His given name was Charles Lutwidge Dodgson, and he taught mathematics at Christ Church College, Oxford. As Lewis Carroll, he not only created a fabulous Wonderland for Alice and wrote nonsense humour, but also set a series of puzzles for his readers, who would write back with the solutions. Much like what we do in Problematics today.

Lewis Carroll, born Charles Lutwidge Dodgson, was a mathematician who engaged readers with puzzles (Source: Public Domain)
Lewis Carroll, born Charles Lutwidge Dodgson, was a mathematician who engaged readers with puzzles (Source: Public Domain)

As a tribute, I have adapted one of Carroll’s best puzzles from that series. The usual issue remains, though: Carroll is so famous that his puzzles abound on the Internet. All we can do is situate the puzzle in a different setting and change the numbers wherever possible. Beyond that, the mathematical principles for solving the puzzle must remain the same as in the original version. So, please try not to look up Carroll’s puzzles before solving my adaptation first.

#Puzzle 75.1


A pair of peripheral expressways in Haryana, famously known as WPE and KMP, form a large ring just outside of Delhi. Buses are allowed to ply here for a toll, but I am not sure if there is any bus service that completes the entire circuit. Probably not, but that doesn’t matter in the following work of fiction.

At a certain point on the twin expressway, which we shall leave unnamed, a bus terminus has been set up. It works with remarkable synchronisation, sending buses in both directions. Every 20 minutes, one bus from either direction reaches that point, halts for a few seconds, and then continues on its journey.

Standing at the point one morning, you and I observe that the buses travelling clockwise are slower than those running anticlockwise. At 10:40am, we note the numbers of both buses that have arrived at that point, just before they set off again. The idea is to record how long it will take each bus to complete one full circuit.

Indeed, the anticlockwise clock reappears first, reaching us at 1:20 pm. By the time the clockwise bus reappears, it’s 2:40pm. Which means, we deduce, that the anticlockwise buses complete the circuit in just 2 hours 40 minutes while the clockwise ones take as long as 4 hours.

“Is there a puzzle in all this?” you wonder, hopefully.

“Indeed,” I reply, drawing from Lewis Carroll. “When the next pair of buses arrive at 3 pm, suppose you board the anticlockwise bus and I board the one going clockwise. Each of us completes the full circuit at their respective speeds. Along the way, we count every bus that we meet travelling from the other direction. We don’t count each other’s bus, because that’s 0 on the number line. Nor do we count the last bus, i.e. the bus that we meet at the end of the journey when we return to this point.”

“Okay,” you agree, “let us board those buses.”

“No, we don’t,” I correct you, “I said ‘suppose’. Without actually boarding the buses, and following the conditions stated above, can you work out how many buses each of us will meet along our respective circuits?”

Can you work it out?


#Puzzle 75.2


Carroll presented his puzzle in two parts, but I am skipping the second part because it’s essentially more of the same. For variety, try this simple one:

The Gregorian calendar that we follow today is an improvement on the Julian calendar, which Caesar introduced in 46 BC. A tortoise born in 46 BC lives an average lifespan (average for tortoises, that is) and dies, coincidentally, on her birthday in 46 AD.

What was the tortoise’s age when she died?




#Puzzle 74.1

Hi Kabir,

Your bookshelf is interesting. You seem to be fond of science fiction.

Say, the number of new books = x

When arranged vertically, books accommodated = x – 50

When arranged horizontally, books that can be accommodated = x + 150

x + 150 = 3 (x – 50)

Solving, x = 150.

— Dr Sunita Gupta, Delhi


Dear Kabir,

I think you have given a very simple Einstein Puzzle. Let's crack it.

As per ANT-MAN's statement, none of the three actors appears in a film whose title has the same number of letters as the actor’s name. This is acknowledged by one of the other actors, who is acting in DRACULA.

Since ANT-MAN is not acting in DRACULA, he must be acting in HALLOWEEN. This means SPIDERMAN and IRONMAN are acting in DRACULA and SCREAM respectively. The solution is shown in the table.

— Sundarraj C, Bengaluru

Solved both puzzles: Dr Sunita Gupta (Delhi), Sundarraj C (Bengaluru), Shishir Gupta (Indore), Akshay Bakhai (Mumbai), Group Captain RK Shrivastava (retired; Delhi)

Solved #Puzzle 74.1: Ajay Ashok (Mumbai), YK Munjal (Delhi)

Solved #Puzzle 74.2: Professor Anshul Kumar (Delhi), Dr Vivek Jain (Baroda), Shri Ram Aggarwal (Delhi)

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

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