Find the square

Hindustan Times

23-06-2025

I learnt something new and interesting recently. Most puzzle solvers, including you readers of Problematics, probably know this already, but here it is anyway. To check if a number is a perfect square, one way to proceed is to keep adding its digits until you reach a single digit. If it's a square, the ultimate 'digital root' in a single digit must be any one of 1, 4, 7 and 9 — and nothing else. Welcome to Problematics!(Shutterstock)
For example, the digital root of 16 is 1 + 6 = 7, that of 81 is 8 + 1 = 9, that of 49 is 4 + 9 = 13 followed by 1 + 3 = 4, and that of 676 is 6 + 7 + 6 = 19 followed by 1 + 9 = 10 and finally 1. There are, of course, numbers that can have digital roots of 1, 4, 7 or 9 without being perfect squares, for example 70 whose root is 7. So the method may not always tell you if a number is a perfect square, but it will definitely help you rule out many possibilities.
#Puzzle 148.1
A century ago, the English puzzler Henry Ernest Dudeney presented six numbers to his readers:
4784887 2494651 8595087 1385287 9042451 9406087
Three of them can be added to form a perfect square, Dudeney wrote, inviting his readers to identify them.
I managed to find the three numbers using hit and trial, paired with some observations that eliminated one or more possibilities. Then I looked at Dudeney's solution and learnt about digital roots and how the rule could be applied to solve this puzzle. Who knows, there may be other methods too. Use any method — hit and trial, digital roots, or any procedure you may innovate yourself. But do send me the three numbers that satisfy the condition.
#Puzzle 148.2
The Village of Music is organising a concert and has invited a musician from the Village of Astronomy. The Village of Astronomy, on the other hand, is inaugurating a telescope and has invited an astronomer from the Village of Music. By sheer coincidence, both events begin at the same time on the same day.
The musician from Astronomy, who wants to get some exercise, decides to walk the whole distance and sets off towards Music at 12 noon on the day of the event. The astronomer from Music chooses to cycle the distance and sets off towards Astronomy at 2pm. Both are good puzzlers and can manage their respective speeds precisely. Indeed, each one arrives at his destination at the exact moment that his event starts.
This is after the two had met on the way, which happened at 4:05pm. They called out 'HI!' to each other but neither stopped. At what time do the two events start?
MAILBOX: LAST WEEK'S SOLVERS
#Puzzle 147.1 HT picture
Hi Kabir,
Based on the conditions that surnames and addresses do not begin with the same letter, and that of four statements only one is true, we can map different scenarios as shown in the table. Statement 1 is not possible. From the other three statements, whichever one is true, only one address can be established for sure, i.e the Poonawala family lives in Surat.
— Sabornee Jana, Mumbai
***
Hi,
The only address that can be definitely established is that the Poonawala family is from Surat. Statement (1) must be false. Statements (2), (3) and (4) give different combinations that do not violate the given conditions. Therefore any two of them can be false. But no matter which one statement is true, one address is common to each combination — the Poonawala family must live in Surat.
— Ajay Ashok, Delhi
#Puzzle 147.2
Hi Kabir,
Each of the ten statements contradicts every other statement. Therefore, at the most one can be true. That is, either exactly one statement is true (this is what statement 9 is) or none of them is true (this is what statement 10 is). However, statement 10 contradicts itself. Therefore, only statement 9 is true and the remaining nine are false.
— Professor Anshul Kumar, Delhi
Solved both puzzles: Sabornee Jana (Mumbai), Ajay Ashok (Delhi), Professor Anshul Kumar (Delhi), Dr Sunita Gupta (Delhi), Yadvendra Somra (Sonipat), Vinod Mahajan (Delhi), Abhinav Mital (Singapore), Shri Ram Aggarwal (Delhi)
Solved Puzzle 147.2: Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Dr Vivek Jain (Baroda) Problematics will be back next week. Please send in your replies by Friday noon to [e-mail]