Problematics | Dickens on the shelves

In the very first edition of Problematics, which I published in a local newspaper in my hometown in 1993, the inaugural puzzle involved three variables and two equations to work with. Now everyone knows, of course, that fewer equations than variables cannot determine a unique set of values. But as puzzlers, we also know that we can work with those equations to derive more equations. That was the basis of my first puzzle—and it is also the basis of my latest one below.

#Puzzle 210.1

A bookstore has 89 copies of Oliver Twist, 28 copies of Great Expectations, and 41 copies of David Copperfield. They are all paperbacks from the same publisher. This means that each copy of any given book has the same number of pages. The storekeeper arranges the books across her shelves, each of which can accommodate about 20,000 pages.

On one shelf, she places 7 copies of Oliver Twist, 11 copies of Great Expectations and 10 copies of David Copperfield, making a total of 19,580 pages, not counting the paperback covers. On the second shelf, she places 13 copies of Oliver Twist, 8 copies of Great Expectations and 9 copies of David Copperfield. This turns out to be an even tighter arrangement, with 19,850 pages accommodated (again, not including the covers).

The storekeeper tries to split the remaining books among three shelves, but finds that they total a little more than 60,000 pages. She solves the problem by placing one copy of each title on display in the store window.

In the days when The Illustrated Weekly of India was still in print, Mukul Sharma published the following puzzle in his column Mindsport.

10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31, 100, 1000, __, 1111111111111111.

The series has 16 numbers (one of which is left blank), and the last number has the digit 1 appearing 16 times.

For a change, we begin with the answer to the second puzzle first, because the first puzzle requires a bit of discussion.

#Puzzle 209.2

Hi Kabir,

The probability that both removed cards are the same colour is 25/51. After the first card is removed, 25 cards of the same colour remain among the unselected 51 cards.

— Shishir Gupta, Indore

#Puzzle 209.1

Hi Kabir,

It looks like there is no guaranteed winning strategy. Here are my thoughts.

Suppose A, B and C are the three players who play in the sequence A B C A B C and so on. The objective of A would be to play such that C defaults so that he/she gets to start the second round. Similarly, B would try to make A default and C would try to make B default. This means that player A would like to form chains of 5, 8, 11… state names. The next best thing for A could perhaps be chains of size 4, 7, 10… which cause B to default. Of course, the worst thing would be if a chain ends after 3, 6 or 9 names, leading to A’s exit.

After considering various alternatives, I could not identify A’s moves that would guarantee C’s exit. There are, however, multiple scenarios where A’s moves can guarantee B’s exit.

— Professor Anshul Kumar, New Delhi

In fact, many readers have sent moves in which A eliminates B first, which is risky as a strategy. Some players use Delhi as the spelling, while Yadvendra Somra uses NCT of Delhi. Let us assume that either spelling is acceptable, but if any one spelling is used then the game cannot name the same Delhi/NCT of Delhi again using the other spelling. Yadvendra has sent a chain in which the first player eliminates the third, but it is based on an assumption.

In the first round, Player #1: Rajasthan, #2: Jammu & Kashmir or NCT of Delhi, #3: eliminated. In the second round, Player #1: NCT of Delhi or Jammu & Kashmir, #2: eliminated.

Yadvendra assumes that in the first round, player #2 does not say Nagaland after player #1 says Rajasthan, because then it gives player #3 the option of Dadra & Nagar Haveli and Daman & Diu. This, in turn, gives various options to player #1.

I disagree with this assumption, and similar ones made by other readers. In Yadvendra’s example, if I were player #2, I would want player #1 to exit first, and would want player #3 to survive until then. In fact, my strategy would be to force player #3 to name a state that would eliminate #1. Player #3 would not want that; he or she would want to force player #1 to name a state that would eliminate me (#2). In those circumstances, after Nagaland from player #1, the optimal move for player #2 is to say Nagaland and let the game continue.

Sabornee Jana makes a similar assumption. The first player says Lakshadweep, and the second player says Puducherry to eliminate the third. This is good for the first player’s point of view. But would the second player want that? In my view, to strategically protect the third player, the second player will prefer to reply to Lakshadweep by naming Punjab or West Bengal.

Since we don’t know any chain of words that aligns with the correct strategy, there are no right or wrong answers. Thank you to everyone who has shared his or her thoughts

Solved both puzzles: Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat), Sabornee Jana (Mumbai), Shishir Gupta (Indore), Dr Sunita Gupta (Delhi), Vinod Mahajan (Delhi)

Solved #Puzzle 209.2: YK Munjal (Delhi), Dr Vivek Jain (Baroda), Ajay Ashok (Delhi), Shri Ram Aggarwal (Delhi)

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