How long does it take to set up an Xmas tree?

This should ideally have appeared last week, but here is a Christmas puzzle I found tucked away in my personal collection. But first, a little background.

Back in 1993, when I was writing a weekly column for a small newspaper in my hometown, a reader, Pramod Pathak, sent me a puzzle and offered it for publication. We didn’t use email in those days and I have long misplaced Pramod’s letter, but I do have a nearly 30-year-old clipping of the column in which his puzzle was published.

Nostalgia aside, I felt the puzzle would work better if I adapted it for readers who are three decades younger. I have changed the form of the original but retained the content.

Belated wishes for Christmas, and have a very happy new year.

#Puzzle 19.1

Having ordered a largish Christmas tree, I found that setting it up was not as easy as I had thought. After struggling for several minutes, I looked around for help and found that the three delivery men were still hanging around, having anticipated that there was money to be earned.

“How much for setting it up?”

“ ₹2,000 if the three of us work together,” they replied in unison. “ ₹1,000 if you engage any one of us, but that will take you a longer time.”

“How much longer?”

“If I work alone, one hour longer compared to the time taken by the full team,” said the man from Andhra Pradesh.

“Five hours longer than the Andhra man if I work alone,” said the man from Bihar.

“Working alone would take me twice as much time as it would take the full team,” said the one from Chhattisgarh.

It’s better to spend ₹1,000 rather than ₹2,000, I thought. That would mean losing some time, but I carefully chose the fastest of the three, given that each of them would charge me the same fee irrespective of his speed.

Which worker did I choose, and how much time did he take to set up my Christmas tree?

#Puzzle 19.2

Take an old-fashioned scale balance and three weights of 1 kg, 2 kg and 5 kg. If you want exactly 3 kg of vegetables, you can either balance the veggies against (1 kg + 2 kg), or you can balance 5 kg against (veggies + 2 kg). That way, you can measure out any weight in integer values from 1 kg to 8 kg.

I have four weights, which I can use to exactly measure any weight from 1 kg to 40 kg (integer values only). What are these weights?

Mailbox: Last week’s solvers

#Puzzle 18.1

Hi Kabir,

Consider 2000 couples. Since the probabilities of having a son and of having a daughter are equal, we can assume that 1000 girls and 1000 boys are born. The 1000 couples who have a daughter can go for a second child, and 500 girls and 500 boys can take birth. And so on.

We can see that number of sons and daughters remain equal.

So, this rule can't ensure the grand objective of having more daughters than sons.

— Madhuri Patwardhan, Thane

#Puzzle 18.2

When we load something on the boat, the boat will displace water, and the water level will thus rise. The weight of the displaced water is equal to the weight of the contents.

When we put these contents directly into the water, it will displace water equal to the volume of the contents.

Now specific gravity (SG) of the contents comes into play. If SG is higher than 1, then less water will be displaced in the second case, and the water level will actually fall.

— Naresh Dhillon, Gurgaon

Naresh also explains what happens when the objects are lighter than water or equally heavy (specific gravity 1), but we can safely assume that garbage (excluding plastics and paper) is usually heavier than water. Amardeep Singh assumes that garbage is lighter than water, but his reasoning is otherwise correct.

Solved both puzzles: Naresh Dhillon (Gurgaon), Aarika Goel (Gurgaon), Gopal Menon (Mumbai), Madhuri Patwardhan (Thane), Pragya Khurana (Ludhiana), Anil Kumar Giyal (Delh9), Amardeep Singh (Meerut), Shishir Gupta (Indore), Rahul Agarwal (Bay Area, California)

Solved #Puzzle 18.1: Abhishek Aggarwal, Narendra Gala (Mumbai), Sanjay Gupta (Delhi), Avneesh Tomar (Delhi)

Solved #Puzzle 18.2: Rohan Rai

Problematics will be back next week. Please end in your replies to problematics@hindustantimes.com.