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Problematics | Books on a library shelf

Jan 13, 2025 02:26 PM IST

Hundreds of books on seven library shelves, unevenly distributed before being shared equally. How many were on each shelf in the beginning?

There are puzzles with multiple solutions, and there are puzzles with a unique solution that can be determined in multiple different ways. The English puzzler Henry Ernest Dudeney (1857-1930) once presented a gambling puzzle and discussed at least two ways to arrive at the solution. I can think of at least one more method: the obvious linear equations.

Representational image.(Shutterstock)
Representational image.(Shutterstock)

However you approach it, the puzzle is of a classical variety: not so simple as to be unrewarding, not so tough as to be unsolvable, and guaranteed fun for however many minutes you require to solve it. To add a touch of originality, I have done away with Dudeney’s gambling background and converted this into a puzzle involving library books.

#Puzzle 125.1

One evening, a librarian gives her apprentice 896 books, returned during the day, to arrange on an empty rack with seven shelves. The apprentice, more interested in literature than arithmetic, arranges them by subject rather than numerically.

In the morning, the senior throws a fit: she wants the books equally distributed across the seven shelves. The junior starts by removing a chunk of books from the top shelf, which has the highest number. She redistributes these among the other shelves, simply doubling the number of books on each. That is to say, if shelf #2 has 27 books, she adds another 27 taken from shelf #1; if shelf #3 has 32, she adds another 32.

It would require a remarkable coincidence for such a single step to bring about an equal distribution, and indeed it does not happen. The apprentice begins again, this time taking a chunk of books from the second shelf. Once again, she takes a chunk of books, this time from the second shelf, and redistributes these among the other six shelves: doubling the number on shelf #1, then doubling the count on shelf #3, and so on all the way down to shelf #7. And once again, she fails to bring about equality.

So she takes a chunk of books out of shelf #3 and goes through the same procedure again. Then shelf #4, shelf #5, shelf #6 and finally shelf #7. Finally, once the books taken out of the seventh and final shelf have been redistributed, the job is done: every shelf has 128 books.

How many books were on each shelf during the night?

#Puzzle 125.2

Puzzle 125.2.
Puzzle 125.2.

The calendars for 2014 and 2025 are exactly the same, as shown in the illustrations for the months of January and February for both years. Over the course of time, you will come across several such duplicate calendars. Suppose you want to create a set of calendars that you and future generations can use and reuse no matter which year it is, going back to any duplicate calendar whenever necessary.

What is the minimum number of different calendars you need to create?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 124.1

Puzzle 124.1.
Puzzle 124.1.

Dear Kabir Sir,

This was my first attempt at a Problematics puzzle. I had a good time solving it. My answer for Puzzle #124.1 is shown in the table.

— Aishwarya Rajarathinam, Coimbatore

#Puzzle 124.2

Puzzle 124.2.
Puzzle 124.2.

Hello Kabir,

This problem can have several answers. The question itself talks about “children”, signifying more than one. If 700 is to be split, between sock type A and B, there can be 6 possible splits which give the required results (plus two splits the results of which cannot be logically divided). The number of children can vary between 2 (at the minimum) to 21 depending on how much money was spent on which sock type.

— Sanjay Gupta, Delhi

The solutions sent by Sanjay Gupta (mail above) and Professor Anshul Kumar (see table) are more or less the same, with tables containing the full list of possibilities. Different readers have sent different numbers of possible answers; everyone is being included in the lists below as long as these possibilities are correct.

Solved both puzzles: Sanjay Gupta (Delhi), Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Geetansha Gera (Faridabad), Shri Ram Aggarwal (Delhi), Shishir Gupta (Indore), Ajay Ashok (Delhi), YK Munjal (Delhi)

Solved Puzzle #125.1: Aishwarya Rajarathinam (Coimbatore), Anil Khanna (Ghaziabad), Rituparna Gupta

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

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