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Problematics | The sum of all house numbers

Oct 21, 2024 12:04 PM IST

Here is an unusual puzzle in which you determine the number of houses on a street without being given any figures. Can you work it out?

The first time I set puzzles for readers was in April 1993, launching a column that was called Problematics (what else?) in a newspaper in my hometown, but my collection of puzzles began much earlier, right from my student days. Whenever I scan my books for a puzzle I can adapt for my present readers, something new pops up every now and then: an unusual or obscure puzzle, or a deceptively simple one that I have not discussed with readers before.

Representational Image.
Representational Image.

The following puzzle is one of them. I think it is medium-level difficult, and you may agree or not, but it is pretty unusual either way. I hope it is the first time you have come across it.

#Puzzle 113.1

Two young siblings stand on the middle of a long road that is lined with many houses. Needless to say, they are numbered serially from #1 upwards.

“What do you reckon is the sum of all these house numbers?” wonders the boy. His sister enlightens him, wisely: “The only way to find out is to add all the numbers.”

Neither of them is old enough to know that there is a formula for the summation of a series of numbers separated by a common difference. So they decide to perform the addition in real time.

“Let’s start adding the numbers, house by house, after dividing the houses between us. You start from one end, and I shall start from the other,” says the girl.

“Which end is mine?” her brother wants to know.

“You are younger, so you start from house #1 and keep moving up. I shall go to the far end, start from the last house (the largest number) and keep moving down,” the girl says. “When we meet somewhere in the middle, we add our respective totals to get the grand total.”

They set off and start adding their planned. Eventually, they meet and compare notes. To their surprise, they find that both have the same subtotal.

“I hope you counted this one, because I didn’t,” says the brother, pointing at the house outside which they have met. “Yes, I did, this house is part of my subtotal,” the sister confirms there is no overlap.

Outside which house have they met, and what is the total number of houses (which is in three digits)?

#Puzzle 113.2

If you found the first one easy, the next one is probably even easier. A ship takes 5 hours to travel from Port A to Port B downstream without halting. The return journey (upstream) at the same speed takes 7 hours. The speed of the river, of course, is constant.

How long will a free-floating raft take to travel from Port A to Port B?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 112.1

Solution to Puzzle 112.1.
Solution to Puzzle 112.1.

Anil Khanna’s table represents the coins as 1 (smallest) to 4 (largest). In any stack of coins, they are marked from the topmost coin down. As the table shows, the puzzle can be solved in 15 steps. Professor Anshul Kumar gives the general formula below:

Hi Kabir,

This is the well-known "Towers of Hanoi" problem. In general, moving a "tower" of n coins from the starting location to the destination involves first moving a "sub-tower" of the top (n – 1) coins from the starting location to an intermediate location. This is followed by moving the nth coin from the starting location to the destination, and then bringing that "sub-tower" back from the intermediate location to the destination. The general formula for the steps required to move n coins is 2ⁿ – 1. Here n is 4, so the number of steps required is 2⁴ – 1 = 15.

— Professor Anshul Kumar, New Delhi

#Puzzle 112.2

Hi Kabir,

I have never quite been an avid Hollywood movie watcher, and the 1940s romcom classic was not something I could recall from memory. So I Googled 1940s Oscar-winning romcom and the names of the actors featured in “The Philadelphia Story" seemed to fit the anagrams. May not be the ideal way to solve this, but the actors are: CARY GRANT, KATHARINE HEPBURN, and JAMES STEWART.

— Sabornee Jana, Mumbai

For the coins puzzle, a couple of readers have taken more than 15 steps. They too are listed among the correct solvers.

Solved both puzzles: Professor Anshul Kumar (Delhi), Sabornee Jana (Mumbai), Dr Sunita Gupta (Delhi), Ajay Ashok (Delhi), Shishir Gupta (Indore), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), Shri Ram Aggarwal (Delhi)

Solved #Puzzle 112.1: Anil Khanna (Ghaziabad), Yadvendra Somra (Sonipat), YK Munjal (Delhi)

Solved #Puzzle 112.2: 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.

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