Problematics | Dining couples

It was only two weeks ago that we met the couples Chatterjee and Das and tried to work out their professions from the way they were seated at a dinner table. Here they are back again, this time out on a holiday.

Some readers may find it unnecessary, but I am dropping a hint. There may be more than one possible arrangement for the people described. You need to examine all these possibilities before answering the last question; that solution is unique across all arrangements.

#Puzzle 132.1

It was only two weeks ago that we met the couples Chatterjee and Das at a dinner table, working out their professions. Here they are back again, this time out on a holiday, where they agree to split the cost of the meals. Breakfast is served to each couple’s room, and therefore is part of each couple’s bill. The four, however, eat lunch and dinner together over the course of their weeklong holiday, and one or the other pays the bill by some kind of rotation.

(1) On Monday, the first day of the holiday, Mr Chatterjee pays for lunch and Ms Chatterjee for dinner.

2) The lunch bills from Tuesday to Sunday (the last day) are paid by Mr Das, Ms Das, Mr Chatterjee, Ms Chatterjee, Mr Das and Ms Das, in that order.

(3) No individual pays for lunch and dinner on the same day.

(4) No individual pays three bills on three consecutive days.

(5) By the end of the week, Mr Chatterjee, Mr Das and Ms Chatterjee have paid four bills each.

(6) One of the above three never pays a bill on the same day that Ms Das pays the other bill. It is some kind of superstitious belief shared by the four.

Which one among the four never pays a bill on the same day as Ms Das?

#Puzzle 132.2

A simpler one. A hotel numbers its rooms in the format FFRR, the first two digits standing for the floor. An ageing professor checks in, and goes out for a walk.

At the hotel, meanwhile, a friend comes to meet the person at the front desk and wants to take her out. “Let me ask boss,” she says. “Will you please stand behind the counter while I am away? Whenever a guest booked into any room comes back, Just hand over the correct key from the panel on the wall.”

It is while she is talking to her boss that the professor arrives from his walk. He finds the young woman’s friend and asks: “Key, please.”

“Room number, sir?”

The professor has forgotten the number. So he asks the young man to check from the database. The red-faced fellow lies: “Sorry, sir, I have just joined and don’t have access to those details.”

The old man tries to remember what he can: “The four digits were all different. The first digit was 1.5 times the last. The difference between the other two digits was 1. The two-digit number representing the room was twice the two-digit number representing the floor.”

“Hey mister,” says the perplexed youth, “if you are so smart, why don’t you work out the number from your own details?”

“Indeed I can,” says the professor, “and the number is ____”

But the young man is hardly listening. His friend has returned from the boss and one look at her dejected face tells him that she hasn’t been allowed to go out in the middle of work.

So, what is the mathematician's room number?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 131.1

Hi Kabir,

NO DEAL. Since the counting was started as a shot was fired, the total shots fired at the end of 60 seconds should have been 61 at the rate of 1 shot/second. But the actual number was 60.

— Sabornee Jana, Mumbai

#Puzzle 131.2

Let C, F and M respectively be the number of marbles initially taken by the child, her father and her mother. From the given statements, we have three equations:

C = 1/5(C + F + M)

C + F = 3M

200 + M = 2(C + F – 200)

Solving, we get C = 96, M = 120, F = 264. Verifying the conditions confirms the solution is correct. The initial distribution was: child 96 marbles, father 264 marbles, mother 120 marbles.

— Vinod Mahajan, Delhi

Solved both puzzles: Sabornee Jana (Mumbai), Vinod Mahajan (Delhi), Dr Sunita Gupta (Delhi), Yadvendra Somra (Sonipat), Anil Khanna (Ghaziabad), Sanjay Gupta (Delhi), YK Munjal (Delhi), Ajay Ashok (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Professor Anshul Kumar (Delhi), Aishwarya Rajarathinam (Coimbatore)

Solved #Puzzle 131.2: Shishir Gupta (Indore), Biren Parmar (Bay Area, California), Dr Vivek Jain (Baroda)

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