Problematics | The rate of eating a growing food course - Hindustan Times

Problematics | The rate of eating a growing food course

Jul 03, 2023 04:27 PM IST

The grass keeps growing on a field as cows keep eating it. How many cows can finish it all up in how many days?

The past week has been one of the busiest for Problematics so far, with over 100 readers attempting my 44th pair of puzzles. Thank you for your response, and keep it going.

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

Almost everyone has solved at least one puzzle, and half have got both correct. Quite a few readers, however, have got the first puzzle wrong because they approached it with a wrong assumption. This is discussed in the Mailbox section.

Meanwhile, here’s another puzzle where a wrong assumption could lead you astray.

#Puzzle 45.1
#Puzzle 45.1

#Puzzle 45.1

Starting today, 30 cows can eat up all the grass in a field in 60 days. Instead of 30, if 70 cows start eating today, they can finish the grass in 24 days. Each cow eats the same amount of grass in a day.

This may look like a mistake. If 30 cows take 60 days to eat the entire field, one cow should take 1,800 days; and if 70 cows take 24 days to do so, one cow should take 1,680 days. How can both be possible if each cow eats the same amount each day?

It’s actually possible if you do not assume that the amount of grass in the field over 60 days is the same as the amount over 24 days. The grass grows at a constant rate until the last blade gets eaten.

How many cows, starting today, can finish the field in 96 days? The source of this puzzle is a century-old Russian book, which I shall name next week.

#Puzzle 45.2

The distance from Delhi to Agra is 240 km. A four-wheeler, carrying an extra tyre, uses all five tyres by rotation over this journey.

What is the distance covered by each tyre if this is equal for all tyres?


Mailbox: Last week’s solvers

#Puzzle 44.1

In last week’s puzzle with candles, the wrong assumption that some readers made was that both candles should burn at the same rate. This cannot be true. Note that the two candles are the same length at 8 am, but this remaining length burns out faster in one candle than in the other. It should be obvious, therefore, that they burn at different rates. Here is one way to solve it:

Hi Kabir,

It takes 18 hours (8 pm to 2 pm the following day) for the longer candle to burn itself out, and 12 hours (12:30 am to 12:30 pm) for the shorter one.

At 8 am, both candles are the same length (say L). To burn this remaining length, the longer candle takes 6 hours (8 am to 2 pm) and the shorter one takes 4.5 hours (8 am to 12:30 pm).

Thus, the original height of the longer candle is (18/6)L = 3L, while that of the shorter one is (12/4.5)L. Their difference is 3 inches, which gives L = 9 inches.

Height of the longer candle = 3*9 = 27 inches, and that of the shorter one = 24 inches.

It is a good mental exercise for a retired person of 70 years like me. Keep sending them.

— Anil Khanna, Ghaziabad

#Puzzle 44.2

This puzzle can have a number of different solutions.

#Puzzle 44.2
#Puzzle 44.2

Solved both puzzles: Anil Khanna (Ghaziabad), Kanwarjit Singh (Delhi), Geeta Arora (Delhi), Dipali Gupta (Delhi), Siddharth Oswal (Ludhiana), Yadvendra Somra (Sonipat), Abhishek Garg (Chandigarh), Dr Sunita Gupta (Delhi), Sunita & Naresh Dhillon (Gurgaon), Mayobhav Pathak (Gurgaon), Ruchir Kohli (Delhi), Abhishek Santra (Faridabad), Amardeep Singh (Meerut), Rohit Khanna (Noida), Shruti M Sethi (Ludhiana), Samarth Girotra (Noida), Shreyas Pathrabe (Navi Mumbai), Subrata Chakravorty (Gurgaon), Subhash C Khanna (Gurgaon), Mudit Singhal (Mumbai), Aidan Williams (Mumbai), Akshay Bakhai (Mumbai), Prof Anshul Kumar (Delhi), Shishir Gupta (Indore), Gaurav Gupta (Indirapuram), Akshita Bhatnagar (Delhi), Madhuri Patwardhan (Thane), Krishna Deshmukh (Mumbai), Hurditya Dand (Mumbai), Sanjay Gupta (Delhi), Ajay Ashok (Mumbai), Anjali Kashikar (Mumbai), Shri Ram Aggarwal (Delhi), Vivek Aggarwal (Bangalore), Umesh CS Bisht (Noida), Anil Goyal (DelhI), Lucky Singh Randhawa & Raman Chaudhary (Mandawali, Delhi), Rajender Prasad Agarwal (Delhi), Sabornee Jana (Mumbai), Joydeep Mahato, Pankaj Sharma, Narendra Prasad, Y K Munjal, Prashant Aggarwal

Solved #Puzzle 44.1: Sandeep Bhateja (Hoshiarpur), SH Idrisi (Delhi), Prakash Bhate (Mumbai), Mansi Gupta (Delhi), Saarth Karkera

Solved #Puzzle 44.2: Parth Gupta (Thane), Nandita Singh (Kalyan), Ashok Kesari (Ghaziabad), Aarika Goel (Gurgaon), Rajesh Bansal (Noida), DP Singh (Delhi), Praveen Kumar Jain (Noida), Aarav Negi (Delhi), Sudip Kumar Behera (Ganjam), Dr Nakul Makkar (Noida), Mayasker Pathak (Gurgaon), Jasleen Kaur (Delhi), Shailja Khosla (Delhi), Vikas Nanda (Pathankot), Madhav Gupta (Delhi), Amar Lal Miglani (Mohali), Kartikeya Rastogi (Noida), Aarush Singhal (Chandigarh), Nathan Sequeira (Mumbai), Shankar Subramanian (Mumbai), Parth Gupta (Thane), Dr Vivek Jain (Baroda), Bisakha (Delhi), Shrikanth Thotapally (Hyderabad), Adeep Atey (Kalyan), Meher Sandhu (Gurgaon), Urvashi Jha (Sarvodaya Kanya Vidyalaya, Mandawali, Delhi), Mukesh Arora (Gurgaon), Guhaan Mehta (Class 9), Nachiket Khare, Tanvee Ghai, Rakesh Yadav, Sunil Gupta, Koel Rai, Dalip Kumar, Manik, Parth Koul, Raja Sharma, Jyoti, Piyush Raina, Hiket Shah, Swarnim Jakhar, Parth Aggarwal, Vaibhav Murade, Yash Chillar, Pooja Thakkar & Nipun Kumar

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

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