Problematics | Think of a number, add several more - Hindustan Times
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Problematics | Think of a number, add several more

Oct 30, 2023 01:48 PM IST

Take any two numbers and keep adding them until you have 10 numbers. What is the quick way to determine the sum of all 10 numbers?

We can multiply any number by 10, 2, or even 5 in our mind, and multiplication by 3, 4 or 6 is not a difficult mental exercise either. Speed calculation is possible with various other multipliers, too, if you know certain rules.

Welcome to Problematics!(Shutterstock)
Welcome to Problematics!(Shutterstock)

The reason for bringing this up is that one of this week’s two puzzles involves mental multiplication by a certain number, but I cannot spell out which number that is. What you may assume, therefore, is that your challenger, described below, knows how to perform speed multiplication in his mind.

#Puzzle 62.1
#Puzzle 62.1

#Puzzle 62.1

Our old friend, the smart alec who plays tricks on you at every party, throws the following new challenge. Turning his back, he asks you to write down any two numbers on a sheet of paper. “Add them up,” he instructs you. So, you now have three numbers.

“Add the second and third numbers,” he continues, and then again: “Next, add the third and fourth numbers,” and, “Now, add the fourth and fifth.”

This goes on. When you have ten numbers on the sheet, the smart alec finally asks you to stop. Turning around, he announces: “Now I shall give you the sum of all 10 numbers.”

He looks at the sheet and, quick as a flash, writes down the sum below the last number.

The numbers shown in the illustration are just an example. The trick can be executed with any pair of starting numbers.

What mental multiplication did the smart alec perform to arrive at the grand total, and why does it work each time?

#Puzzle 62.2
#Puzzle 62.2

#Puzzle 62.2

In one hourglass, the upper bowl, when full, takes exactly 11 minutes to empty itself into the lower bowl. In the other hourglass, this takes 7 minutes.

Starting with both upper bowls full, how would you measure exactly 15 minutes? You may invert either or both hourglasses whenever necessary.

Mailbox: Last week’s solvers

#Puzzle 61.1

Kabir Sir,

The word can be either BID or CRY.

BID CRY DRY FIG RIM

BID: Person #1 has B, so he got the word. Person #2 has D, and was unsure between DRY and BID, but from #1’s answer, he got the word. Person #3 has I, and was unsure among BID, FIG and RIM. But FIG was not the word as both #1 and #2 would have replied immediately. Again, if #1 had M, #2 would have been undecided between CRY and RIM. So, #3 understood that #1 must have B and #2 must have D.

CRY: Person #1 has C. Person #2 has Y, and was unsure between CRY and DRY, but would know now that #1 does not have R as it appears more than once. So, #1 must have C, from CRY. Person #3 has R, which appears in CRY, DRY and RIM. DRY can be ruled out, because #1 wouldn’t have been able to answer with D or Y. If the word was RIM, #1 would have M but #2 (with I) wouldn’t have been able to guess whether the word was BID or RIM. So, #3 knew that # 1 must have C and #2 must have Y.

— Ayush Tandon, Delhi Technical University

#Puzzle 61.2

Hello Kabir,

If there are S students, out of whom H scored 100, the probability that two students picked randomly both got 100 is:

H/S x (H – 1)/(S – 1)

This is given to be 1/2. In the range 20 to 30, the only whole numbers that match this condition give us 21 students out of whom 15 scored 100.

— Amit Khanna, Fremont, California

Solved both puzzles: Ayush Tandon (DTU), Amit Khanna (Fremont), Bhasker Mundhra (Ghaziabad), Dr GL Arora (Delhi), Yadvendra Somra (Sonipat), Rajesh Bansal (Noida), Prof Anshul Kumar (Delhi), Dr Sunita Gupta (Delhi), Shri Ram Aggarwal (Delhi), Mukund Sharad Hejib (Gurgaon), Vivek Aggarwal (Bangalore), Kanwarjit Singh (Delhi), Akshai Bakhai (Mumbai), Vineet Kumar Dargan (Delhi), Ar Kamal Passi (Central PWD), Group Captain RK Shrivastava (retd; Delhi), Rachna Jain (Delhi), Sunil Kumar Agarwala

Solved #Puzzle 61.1: Aarika Goel (Gurgaon), Shawn Jacob (Mumbai), Anil Kashyap (Panchkula), Abhinav Gupta (Rewari), Bhuvi Jain (Delhi)

Solved #Puzzle 61.2: Vishal Rawat (Ambala), Anil Khanna (Ghaziabad), Shruti (Ludhiana), Ajay Ashok (Mumbai), Sushma Pandhi (Chandigarh), Abhinav Gupta (Haryana), Rohit Sahay (Delhi), Shishir Gupta (Indore), Arun Kumar Gupta (Greater Noida)

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

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