Problematics | The fast and the furious - Hindustan Times

Problematics | The fast and the furious

Feb 05, 2024 11:26 AM IST

We give you three runners, the differences in their speeds and in the time they clocked. What is the length of the race, and how long did each runner clock?

The puzzles in these columns over the last couple of weeks were possibly a little on the tougher side. On both weeks, there have been wrong answers from some readers who usually solve everything correctly. This week, we shall go with a couple that I think you will find somewhat simpler.

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

So, let us return to speed and distance. As past experience in Problematics has shown, these puzzles are not particularly difficult, but they are always fun to work on.

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#Puzzle 76.1

At a local track and field event, a race is about to start but most of the villagers are furious. One of the contestants is a professional runner, which obviously gives him an unfair advantage. The professional, for his part, argues that he belongs to the village, which has all the qualifications he needs. There is nothing in the rules that says a professional athlete cannot compete in this local competition, he points out. In the end, this argument settles the dispute. The village committee allows him to participate while resolving to revise the rules ahead of the next year’s event.

At the end of the race, everyone is complaining again. Everyone, that is, except the professional, who has won as expected. His victory was even more comfortable than he had expected to be. For context, take the contestant who finished second. He was a decent runner himself, completing the race a full 20 seconds earlier than the one who finished last. Yet the champion defeated the second-placed runner by an even wider margin: 30 seconds. Such was the difference in quality.

The champion, as we said, is not complaining. Being a fan of puzzles apart from running, he calculates that he ran 10.8 km/hour faster than the second-placed runner who, in turn, ran 3.6 km/hour faster than the one who finished last.

What was the length of the race, and what timings were clocked by the runners who finished first, second and last?

#Puzzle 76.2

#Puzzle 76.2
#Puzzle 76.2

One way of solving Wordle is by using up to 25 different letters in the first five attempts. It is not a particularly good strategy, because you will require all six attempts, and even that may not be enough if the letters you have unearthed can be arranged into more than one possible solution. Then there is also the possibility of repeating letters.

That said, if you do use this strategy, there are some combinations of five words that use 25 different letters, but some of those words are not common. On the other hand, you could try five common words that use 22, 23 or 24 different letters. In the example pictured, the five words use 24 different letters, with B and Q not represented, and with L appearing twice.

The usual rules apply. The secret word has five letters. In your test words, for each of the five letters, green means that the highlighted letter appears in that same place in the hidden word, while yellow means that this letter appears in a different position in the secret word.

If a player uses these five words and gets results as shown in the image, can they uncover the word with certainty on the sixth attempt? Or is there more than one possible solution?


#Puzzle 75.1

Anticlockwise buses complete the circuit in 2 hours 40 minutes or 160 minutes. There will have to be 8 buses to ensure the 20-minute frequency anticlockwise. Similarly, the clockwise circuit is completed in 4 hours or 240 minutes. There will be 12 buses running clockwise to ensure the frequency of 20 minutes.

Take the person travelling clockwise. When he or she starts, 8 anticlockwise buses are on the circuit, including the one with this person. In addition, in the 4 hours it takes to complete the circuit, 12 more buses will leave the starting point (some buses will be seen twice). This person will see a total of 20 buses. Excluding the buses in which the other person is travelling anticlockwise and the bus he will meet at the starting point, the number of buses seen will be 18.

The person travelling anticlockwise will similarly meet all 12 buses on the clockwise circuit plus 8 more that will leave the starting point in the 160 minutes of his or her travel. Excluding the bus in which the other person is travelling clockwise and the last bus, the total number will be 18 again.

— Kanwarjit Singh, Chief Commissioner of Income Tax (retired)

#Puzzle 75.2

Hi Kabir,

Since the year 1 BC is followed by 1 AD (there is no year 0), the life span of the tortoise from 46 BC to 46 AD is 46 + 46 – 1 = 91 years.

— Professor Anshul Kumar, Delhi


46 BC is 45 years before the start of the AD era. Hence the age of the tortoise when it died was

(45 in BC era) + (46 years in the AD era) = 91 years.

— Yadvendra Somra, Sonipat

A couple of readers took note of the point that Year 0 did not exist but then miscalculated the answer. In contrast, in #Puzzle 75.1, one reader did all the calculations correctly but deleted only one bus instead of two, so ended up with a count of 19 (I am counting this one as correct, though).

Solved both puzzles: Kanwarjit Singh (Chief Commissioner of Income Tax, retired), Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat), Shishir Gupta (Indore)

Solved #Puzzle 75.1: Akshay Bakhai (Mumbai)

Solved #Puzzle 75.2: Dr Sunita Gupta (Delhi), Group Captain RK Shrivastava (retired, Delhi), Sundarraj C (Bengaluru)

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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