In the last century, Royal Vale Heath, a New York stockbroker, invented a smart trick using the serial number of a US dollar bill, which had eight digits during his time. Martin Gardner took Heath’s permission in explaining the trick, and my source is Gardner’s writings.
Since Heath and Gardner are no longer with us, I cannot ask either of them for permission. Not that I am going to explain the trick anyway; I shall only describe it. Why it works will be a puzzle for you to solve.
#Puzzle 176.1
To make Heath's trick more relatable to Problematics readers, who are mostly Indian, I am describing it using telephone numbers. In cities with a three-digit STD code, such as Delhi (011), Mumbai (022) or Kolkata (033), landline phone numbers usually come in 8 digits. Say 011-23456789.
Ask an audience member to note down her landline number without the STD code. Then ask her to add the digits in consecutive pairs (first + second, second + third, third + fourth, fourth + fifth, fifth + sixth, sixth + seventh, and seventh + eighth). As she adds, she calls out all these seven sums, one by one. Finally, ask for an eighth sum: second digit + eighth digit. She gives you that too.
And lo, you announce her entire eight-digit phone number.
Without giving away too much, I can tell you this: while she was calling out the sums, you were doing your own mental arithmetic. You placed four of the eight sums in one group, and three of the remaining four sums in another group. A series of arithmetical steps with these two groups allowed you to determine one of the digits. The rest followed.
How is the trick done?
#Puzzle 176.2
On her 100th birthday, a reader of Problematics in HT (which began in its current version in 2022) discussed a mathematical curiosity with her great-grandchildren, and wondered if it was interesting enough to work as a puzzle for our column.
“In the year I was born, my elder sister was as old as the last two digits of her birth year. Coincidentally, in the same year, our grandmother too was as old as the last two digits of her birth year. The ages of my grandmother and my sister were both even numbers that year,” she told the kids.
“Both Granny and Didi, sadly, died on their respective 100th birthdays,” she continued, “but I hope to be around for a few more years more with your prayers.”
The elderly woman is still with us, cheerful as ever. She celebrated her latest birthday in the first week of the New Year.
How old is the Problematics reader, and in which years did she lose her grandmother and her sister?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 175.1
Dear Mr Kabir,
I find myself in a bit of a dilemma about the level of this puzzle. One possible interpretation is that, following the standard pattern of the Indian Railway reservation system, berth number 20 would be the lower berth and berth number 21 the middle one. Yet, I feel the puzzle cannot be this simple, given the complexity you've consistently brought to your puzzles over the past three years.
— Shri Ram Aggarwal, Palam, New Delhi
In fact, Mr Aggarwal, it was meant to be simple puzzle this time; a question of using arithmetic to determine the seat orientation, as shown below — Kabir
Hello Kabir,
In the first puzzle, your illustration was a big help. The way to solve it is: take the seat number, divide by 8 and look at the remainder. The following table tells us the location of any seat:
Remainder Seat
0 Side upper
1, 4 Lower
2, 5 Middle
3, 6 Upper
7 Side lower
For seat number 20, the remainder is 4, so it is a lower seat. For number 21, the remainder is 5, so it is a middle seat. Therefore, the sister has a lower berth and the brother has a middle berth.
— Dr Sunita Gupta, New Delhi
#Puzzle 175.2
Hi Kabir,
The berries collected by the children are: Arjun: 38, Bhagat. 34, Charulata: 1296, Dilip: 18, Esha: 36, Farida: 72. As given, Charulata's collection must be the square of the square of an even integer, and the grand total must be between 1000 to 1500. The only number for Charulata that satisfies all the given conditions is 6⁴ = 1296. The collection of the other children can now be calculated, and the grand total is 1464.
Wish you a very Happy New Year. Waiting for more and more exciting puzzles this year.
— Shishir Gupta, Indore
Solved both puzzles: Shri Ram Aggarwal (Delhi), Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), Sabornee Jana (Mumbai), Vinod Mahajan (Delhi), Anil Khanna (Ghaziabad), Professor Anshul Kumar (Delhi), Ajay Ashok (Delhi), Yadvendra Somra (Sonipat), YK Munjal (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Amarpreet (Delhi)
Solved #Puzzle 175.1: Dr Nitin Trasi (Sydney)