Problematics | A checklist of handshakes
10 couples sit together for lunch, each shaking hands a different number of times. Whose handshakes can you work out?
Over decades of puzzling, I have accumulated a number of books from the late American author Martin Gardner. His amazing stock of puzzles never ceases to surprise, and I found the latest in an unexpected source. It’s in a compilation put together by another writer, crediting Gardner who in turn credits one of his readers. As far as I can recall, this puzzle does not appear in any of the Gardner books in my collection.
A quick internet check showed the puzzle does have a following. Someone has even made a video explaining the solution, but I expect Problematics readers would like to attempt it themselves. The puzzle looks impossible to solve at first, but it can be done.
#Puzzle 158.1
We have invited four couples to lunch. Our daughter is away at work, so there are just the 10 of us: the 8 guests, my wife and myself.
As the guests arrive one by one, the 10 of us exchange handshakes. Not everyone shakes hands with everyone else, however. No one shakes hands with their own spouse, and obviously no one shakes hands with themselves. Some of us shake hands with more people, some with fewer. No one shakes hands with the same person more than once.
I notice these discrepancies and set out to make a checklist, full of optimism that there is a puzzle to be solved. “How many hands did you shake,” I ask one of the guests. She gives me a number. “And you?” to her husband, who gives me another number. I put the same question to the other 6 guests, and finally to my wife.
“Interesting,” I note, looking at the checklist. “Each one of you 9 people has been involved in a different number of handshakes. All 9 numbers are different.”
How many people did my wife shake hands with?
#Puzzle 158.2
Needing to get three rooms painted, Santa Claus engages a team of three. For convenience, and also to protect their true identities, let us call the painters Mickey, Donald and Popeye.
Santa was hoping that each of them would paint one room, but that is not how they work. On the first day, only Mickey and Donald come and paint a single room, finishing it in 8 hours. The second day, Mickey comes with Popeye, and they paint another room, taking 9 hours. On the third day, while Mickey stays home, Donald and Popeye paint the third room in 10 hours.
Each one works at his own speed, which is constant whether it is the first day or the second day or the third. Each room is the same size, requiring the same amount of work.
If each one painted a room alone, how many hours would each one take? The answers are not integer values.
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 157.1
Dear Kabir,
In a standard die the sum of the spots on any pair of opposite faces is always 7. Therefore, when three dice are kept one over the other, the sum of the three pairs of opposite horizontal faces will be 21. The magician keeps 21 matchsticks in his concealed hand. He drops into his pocket as many matchsticks as the number of dots visible on the top face of the top die, takes out the remaining and scatters these on the table.
— Yadvendra Somra, Sonipat
#Puzzle 157.2
Hi Kabir,
I had come across this puzzle in my childhood. Nice to see it coming up again. The sequence of steps that will achieve the desired result is shown below, along with the state reached after every step. The containers with capacities 8L, 5L and 3L are denoted by C₈, C₅ and C₃, respectively and amounts of water in these are denoted by W₈, W₅ and W₃.
Initial state (W₈ = 8, W₅ = 0, W₃ = 0)
1. Transfer 5L from C₈ to C₅ (W8 = 3, W₅ = 5, W₃ = 0)
2. Transfer 3L from C₅ to C₃ (W₈ = 3, W₅ = 2, W₃ = 3)
3. Transfer 3L from C₃ to C₈ (W₈ = 6, W₅ = 2, W₃ = 0)
4. Transfer 2L from C₅ to C₃ (W₈ = 6, W₅ = 0, W₃ = 2)
5. Transfer 5L from C₈ to C₅ (W₈ = 1, W₅ = 5, W₃ = 2)
6. Transfer 1L from C₅ to C₃ (W₈ = 1, W₅ = 4, W₃ = 3)
7. Transfer 3L from C₃ to C₈ (W₈ = 4, W₅ = 4, W₃ = 0)
— Professor Anshul Kumar, New Delhi
Solved both puzzles: Yadvendra Somra (Sonipat), Professor Anshul Kumar (Delhi), Anil Khanna (Ghaziabad), Dr Sunita Gupta (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Shri Ram Aggarwal (Delhi), Dr Vivek Jain (Baroda), Ajay Ashok (Delhi), Sanjay Gupta (Delhi), Vinod Mahajan (Delhi)
Solved #Puzzle 158.1: YK Munjal (Delhi)
Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com.
