Problematics | When entertainment was inexpensive
A puzzle with some ₹100 notes and some ₹1 coins, with only a little spent.
A delightful chestnut included in many puzzle collections, including the writings of great puzzlers such as Henry Dudeney and Martin Gardner, describes a mismatch between the amount written on a cheque and some cash. The transactions involved in the original version cannot work in the modern world, whichever currency one uses, so I have adapted it into an entirely new story with appropriate pricing changes.
#Puzzle 156.1
Decades ago, when no one had heard of UPI payments, a couple going out for a movie had all their money in ₹100 notes and ₹1 coins. The wife had as many ₹100 notes as the husband had ₹1 coins, and she had as many ₹1 coins as he had ₹100 notes. Because she was carrying more money, she paid for the tickets, which cost ₹5 (this was actually possible up to the early 1980s).
“Rich woman,” the husband teased his wife. “Even after spending ₹5, you have twice as much money as I do.”
Who had how much money when they went out?
#Puzzle 156.2
Three professors at a college, two of them men and one a woman, teach three different subjects: physics, botany and mathematics. Chatting at the canteen, the woman tells her two colleagues that she is planning a trip to Punjab, the home state of one of the two men.
Something then strikes the woman. “Hey, the three of us teach subjects that begin with B, M and P,
and our home states — Bihar, Manipur and Punjab — also begin with those three letters!”
The professor from Bihar nods and says, “I noticed that long ago, and also that none of us teaches a subject that begins with the same letter as his or her name.”
The mathematician looks stunned and says: “I had never noticed this before.”
You now have one statement from each of the three.
Which professor is from which state, and which one is the woman?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 155.1
Hello Kabir,
The solution to this puzzle is in the form of a table, as shown.
— Anil Khanna, Ghaziabad
#Puzzle 156.2
Suppose a number has k digits. Its cube has at most 3k digits. Therefore, the sum of the digits of the cube can at most be 27k. The smallest k-digit number is 10k – 1. It can be seen by observation that for all values of k larger than 2, the smallest k-digit number is always larger than the maximum possible sum of the digits of its cube. In other words, the digits of all numbers of 3 or more digits will always have a sum that is greater than their respective cubes. Therefore, we need to consider cubes of 1-digit and 2-digit numbers only. An exhaustive search reveals that only the following numbers (ignoring 0 and 1) have the property that the sum of their digits equals their respective cube roots:
8 = 5 + 1 + 2
17 = 4 + 9 + 1 + 3
18 = 5 + 8 + 3 + 2
26 = 1 + 7 + 5 + 7 + 6
27 = 1 + 9 + 6 + 8 + 3
— Professor Anshul Kumar, New Delhi
***
Hi,
These are known as Dudeney numbers. The other numbers with the same property are 18 (cube root of) 5832, 26 (cube root of 17576) and 27 (cube root of 19683).
— Ajay Ashok, New Delhi
Solved both puzzles: Anil Khanna (Ghaziabad), Professor Anshul Kumar (Delhi), Ajay Ashok (Delhi), Dr Sunita Gupta (Delhi), Vinod Mahajan (Delhi), Shri Ram Aggarwal (Delhi), YK Munjal (Delhi), Yadvendra Somra (Delhi), Shishir Gupta (Indore), Sanjay Gupta (Delhi)
Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com.
