We learn some properties of numbers early in life, such as all products of 9 having a digital root of 9, and all single-digit multiplications with 37 leading to products with three repeating digits. Other properties may pass unnoticed for years until they are presented to us in the face, and we wonder why we hadn’t noticed it all along.
The following puzzle exploits a property I had not noticed until I first read and solved the puzzle some three decades ago. Martin Gardner was particularly fond of it, and credits it to two readers who brought it to his notice separately. My version tweaks the original described by Garden. The puzzle may seem to lack enough information when you first read it, but think a bit and you will find it is as easy as it is enjoyable.
#Puzzle 152.1
A publisher hands a number of copies of a school textbook to two vendors. The deal is that the book is to be sold at its cover price, with the vendors receiving a commission based on the number of copies they sell.
By coincidence, the number of copies sold turns out to match the commission in rupees per book. That is to say, if the commission is x rupees per book, then they have sold exactly x copies. The publisher pays them the commission in ₹10 notes and ₹1 coins, the number of coins being less than 10. Having no other cash in their pockets, the vendors decide to share the notes and coins in the style of Blondie and Tuco in The Good, The Bad and The Ugly.
{{/usCountry}}By coincidence, the number of copies sold turns out to match the commission in rupees per book. That is to say, if the commission is x rupees per book, then they have sold exactly x copies. The publisher pays them the commission in ₹10 notes and ₹1 coins, the number of coins being less than 10. Having no other cash in their pockets, the vendors decide to share the notes and coins in the style of Blondie and Tuco in The Good, The Bad and The Ugly.
{{/usCountry}}“One for you, one for me,” Vendor #1 says, beginning to distribute the ₹10 notes. “One for you, one for me…” And so on, until the last note is reached: “One for you.”
It strikes Vendor #1 that he has handed Vendor#2 the first note as well as the last. “Hey, you got ₹10 more than I did because the number of notes was odd.”
“Never mind,” says Vendor #2, “you keep all the ₹1 coins.”
“But that’s less than 10 coins and so less than ₹10. You still end up with a higher share of our commission,” says Vendor #1.
“Never mind,” Vendor #2 repeats himself. “Let me pay you the difference.” He opens his UPI app, and makes the transaction. “There, we now have equal shares.”
How much does Vendor #2 send Vendor #1 by UPI?
#Puzzle 152.2
A pet shop manager assures a customer that the parrot he is offering will repeat every word she hears. The customer tries to check this out but the parrot is fast asleep, having been drugged by the manager. To make sure he is not cheated, the customer makes the manager put the assurance in writing: “Guaranteed that Parrot #152.2 sold to [customer’s name] in July 2025 will repeat every word she hears, failing which the payment of [amount] shall be returned to the customer.”
Thus assured, the customer buys the parrot. At home, when the pet wakes up, the buyer says “Hi!” No reply. He tries various other words, but not a word from the parrot. The buyer rushes angrily to the pet shop, but the manager refuses a refund claiming no terms have been breached. The buyer goes to the local don for arbitration. To his disappointment, the don rules in the manager’s favour.
Explain why the manager hadn’t bluffed.
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 151.1
Hi Kabir,
Here is how the mathematical trick with cards works. The digital roots of the numbers from 43 to 51 are: (43 = 7), (44 = 8), (45 = 9), (46 = 1), (47 = 2), (48 = 3), (49 = 4), (50 = 5), (51 = 6). We can see that the sequence of digital roots matches the sequence of cards 7-8-9-1-2-3-4-5-6. Whatever the number of cards added to the pile of 43 cards, the new digital root will match the card that is not transferred and turned face up. Also, note that the digital root of any number will remain the same even after splitting that number into two or more parts and adding the digital roots of those parts.
— Shishir Gupta, Indore
#Puzzle 151.2
Hi Kabir,
The four cards are, from left to right, are: Queen of Spades, Ace of Hearts, Jack of Clubs, King of Diamonds.
— Aditya Krishnan, NMIMS Mumbai
Solved both puzzles: Shishir Gupta (Indore), Aditya Krishnan (NMIMS Mumbai), Vinod Mahajan (Delhi), Dr Sunita Gupta (Delhi), Anil Khanna (Ghaziabad), YK Munjal (Delhi),Sanjay Gupta (Delhi), Professor Anshul Kumar (Copenhagen), Ajay Ashok (Delhi), Yadvendra Somra (Sonipat)
Solved Puzzle 151.2: Dr Vivek Jain (Baroda)
Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com.