Problematics | How to complete your course before the exams - Hindustan Times

Problematics | How to complete your course before the exams

Jan 08, 2024 12:29 PM IST

The speed at which you intend to study often ends up as wishful thinking. Meet a student who went a bit slow and figure out how much she needs to make up.

Parents never tire of telling their children that marks in exams were awarded more strictly in their days than they are now. This is true, going by the astronomical marks we used to see in the race for college seats before introducing a common entrance test. What is also true, I think, is that students of today work harder in general (there will always be exceptions) than their counterparts from previous generations.

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

Some things, however, don’t change. For example, that late rush to complete or revise one’s course when the exam dates draw near. Many students would set themselves deadlines, and still do, hoping to complete this part or that part of their course in these many days, but it rarely works out according to that schedule and usually ends with a hectic scramble in the last few days.

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Our puzzle includes one such late rush by a student. It doesn’t matter which generation she belongs to because, as we have already noted, this is one of those things that remain the same from one generation to the next.

#Puzzle 72.1

There’s a week to go before the exams and our student, although a serious one, is far from completing her first revision. She counts the remaining chapters across subjects and sets herself the only possible deadline: all these must be completed within the remaining seven days.

On the first day, she completes only one-tenth of what remains to be revised. That’s below the necessary average of one-seventh per day, she notes nervously. The next day, however, instead of upping her pace, she actually slows down, completing two-thirds of what she had covered the previous day. These are different chapters, of course, just that the number of chapters (which may or may not be an integer) covered on one day is two-thirds of the number covered on the previous day.

On the third day, her speed is still slower than on the first day. She again covers one-tenth of what remains. On the fourth day, she covers two-thirds of the total she has covered so far. And so on, one-tenth of what remains, alternating with two-thirds of what she has already covered.

She does not meet the deadline. At the end of the seventh day, she finds that she still has 15 chapters unrevised. Luckily, those are for papers that are scheduled for later. Determined to raise her speed from now on, she heads for her first paper, which is happily covered in full.

How many chapters remained to be revised when she set herself the deadline with seven days to go?

#Puzzle 72.2

#Puzzle 72.2
#Puzzle 72.2

You have a 4x4 board, on which I have temporarily placed four blue counters. In addition, you also have three red, three orange, three yellow and three green counters. Place all 16 counters on the board, one in each cell, such that no colour appears more than once in any row or column.


#Puzzle 71.1

Hi Kabir,

A very Happy New Year to you.

As I understand, everybody is saying the person next to them is a liar. The only way this can happen when standing In a circle is if the liars and truth speakers are standing alternately. So, there has to be an even no of people, otherwise two liars or two truth-tellers will be next to each other, which will not satisfy the condition.

The guy in the red shirt must be a liar because the number he gave is odd, viz 51. The woman in the blue dress cannot be a liar because she is calling the guy a liar. Hence, she is speaking the truth. So, the number of people should be 54.

— Dr Sunita Gupta, Delhi

The two deductions above — that the number of people must be even, and that the woman is calling the man a liar, and is therefore truthful — are both necessary for solving the puzzle. Some readers have got the first point but missed the second. Without her assertion that the man is a liar, both of them could have been lying and the total number of people could have been any even number.

Again, some have argued that “no one can be a born liar”, as described in the puzzle, and while that is indeed true, it was only intended as a figure of speech. My apologies to readers who read too much into that point.

#Puzzle 71.2

Hi Kabir,

Wish you a Happy New Year.

The original payment of 30,000 made by the tenants, after changing hands, is accounted for as follows: with the landlord: 25,000; with the landlord's son: 2,000; with the 3 tenants: 1,000 each = 3,000

In summary, 30,000 = 25,000 + 2,000 + 3,000

Another way of looking at it is as follows: Effectively, the rent paid by the tenants was 27,000. Out of this, the landlord got only 25,000 and his son pocketed 2,000.

That is, 27,000 = 25,000 + 2,000

— Professor Anshul Kumar, Delhi

Solved both puzzles: Dr Sunita Gupta (Delhi), Prof Anshul Kumar (Delhi), Sundarraj C (Bengaluru)

Solved #Puzzle 71.1: Kanwarjit Singh (Chief Commissioner of Income Tax, retired)

Solved Puzzle 71.2: Yadvendra Somra (Sonipat), Ajay Ashok (Mumbai), Vivek Jain (Mumbai), Shishir Gupta (Indore), Shri Ram Aggarwal (Delhi), Group Captain RK Shrivastava (retired; Delhi), Jaikumar Inder Bhatia & Disha Bhatia (Ulhasnagar, Thane), Abhishek Garg (Chandigarh)

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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