Problematics | Double or nothing

Most people with an interest in puzzles would be familiar with the wheat and chessboard puzzle. In the basic version, you place 1 grain of wheat on the first of the board’s 64 squares, then 2 grains on the second square, 4 on the third, 8 on the fourth, 16 on the fifth and so on, doubling the number of grains on each successive square. By the time you are done with all the squares, the 64th alone would have 9,223,372,036,854,775,808 grains (assuming it could accommodate that many) while the entire chessboard would be covered with as many as 18,446,744,073,709,551,615.

When a Google search leads you to detailed analyses of the solution, the puzzle in its original form cannot find a place in Problematics. Adaptations are, however, fine. The following version, which I found in an old book of puzzles, takes the basic idea from the chessboard puzzle and places it in a different context.

#Puzzle 95.1

It’s the Christmas/New Year season and an unscrupulous man has just read about the chessboard puzzle for the first time. Why not use this idea, he tells himself, to lure my neighbour into a trap? So he goes next door with a proposition: “Would you like to make a bet with me?”

The neighbour, who is understandably suspicious, asks for the details. The conman’s proposition: “Starting New Year’s Day, I shall deposit ₹5 lakh in your account daily for the entire month. In return, you deposit 1p into my account on January 1, 2p on January 2, 4p on January 3, and so on, doubling the amount each day. At the end of the month, each of us stops making these deposits in the other’s account. Deal?”

The neighbour is shrewd. “I shall be away for the New Year. Let us discuss this again after I come back,” she tells the conman.

A few weeks later, the neighbour calls on the conman. “I am back. How about taking up your bet next month, making our payments to each other every day from February 1 to 28?”

Should the conman take the bet this time?

#Puzzle 95.2

In the real world, unfortunately, it is not the Christmas/New Year season. The summer in Delhi, where I live, has been unbearably hot, and many other cities too have been suffering record temperatures. It is in such sweltering conditions that a man drives along a highway, going from village to village as part of his work. It’s fine inside the AC car but the heat hits him whenever he steps out.

The man decides that he will take a haircut in the next village in the hope that a lighter head will reduce his suffering. When he reaches the village, he finds it has two barbers. One wears a neat hairstyle that any customer would desire, while the other barber’s hair is rather unruly.

Which barber should our man visit?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 94.1

Hi Kabir,

There are 8 possibilities considering that each woman either ordered ice cream or didn't. Out of these, only the following one satisfies all the four conditions that are given: The women from Andhra Pradesh and Assam order ice cream and the woman from Arunachal Pradesh does not order ice cream.

— Professor Anshul Kumar, Delhi

Before going to the solution of the next puzzle, a brief discussion on the above one. A number of readers got it wrong, with a few insisting that there could be two possible solutions:

(1) The women from Andhra Pradesh and Assam order ice cream while the woman from Arunachal Pradesh does not (which is correct)

(2) Only the woman from Arunachal Pradesh orders ice cream while the other two do not (which is wrong)

The second option above is not possible because it is ruled out by condition 3 in the puzzle: “The woman from Andhra Pradesh must have ice cream unless neither of her colleagues orders ice cream for themselves.” This means that if any of the other two women orders ice cream, then the Andhra Pradesh too must do so. In other words, neither the woman from Assam nor the woman from Arunachal Pradesh can order ice cream alone.

#Puzzle 94.2

Dear Kabir,

The month can only be 11. Days can have 20 values. Correspondingly there will be some value for the years (8 pre-independence dates, and 12-post independence dates). The details are shown in the table.

— Sampath Kumar V, Coimbatore

I have to say that Sampath has offered possibilities that did not strike me when I presented the puzzle. In fact, like me, most other readers appear to have missed those possibilities: In digital text (as in a calculator), 2 and 5 read the same upside down (besides the obvious 0, 1 and 8, while 6 becomes 9 and 9 becomes 6 upside down).

In the lists below, anyone who has offered at least one correct date is credited.

Solved both puzzles: Professor Anshul Kumar (Delhi), Sampath Kumar V (Coimbatore), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), Akshay Bakhai (Mumbai), Harshit Arora (IIT Delhi)

Solved #Puzzle 94.2: Dr Sunita Gupta (Delhi), Dr Vivek Jain (Baroda), Avanti Kashikar (Mumbai), Sanjay S (Coimbatore), Shishir Gupta (Indore), Ajay Ashok (Mumbai), Jaikumar Inder Bhatia & Disha Bhatia (Ulhasnagar, Thane)

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