Problematics | The uncle and the grandmother
A mathematician has two children, a brother and her mother; determine the ages of all four
One of the joys of writing this column has been to observe the way readers have solved my puzzles in their own innovative ways, which are often different from the way I would have solved them myself. This has happened most frequently with puzzles involving Diophantine equations, for which we seek integral solutions. The fun part, of course, is that the puzzles in Problematics involve fewer equations and more variables, otherwise they would have been plain textbook exercises.
There are no set rules for how you should solve the following puzzle. There never are. You could use a mix of mathematics and reasoning based on observation, or you could simply go with hit and trial. All that matters is that you send me the correct solutions before next week’s column.
#Puzzle 120.1
A mathematician with two young children is showing off her skills to her mother and younger brother.
“First you, Mother,” she says. “Take your age and add it to the age of my son.”
The older woman notes down the sum of her age and her grandson’s.
“Now subtract his age from yours,” the mathematician says. The middle-aged woman notes down the difference.
“Now multiply your ages,” her daughter gives the third step. The granny notes down the product too.
“Finally,” the mathematician says, “divide your age by my son’s.” Her mother does so and notes down the result.
“How many numbers you have noted down?” the younger woman wants to know.
“Four numbers,” replies her mother. “The sum of our ages, the difference between them, their product, and the quotient of my age divided by the boy’s age.”
“What is the total of those four results?”
“243,” the older woman says, “but where is all this leading to?”
“You will soon see,” her daughter says. She now addresses her brother: “Your turn now. Take your age and add it to my daughter's age.”
The young man notes down the sum.
The same steps follow: “Subtract”, “Multiply”, and “Divide”. Finally, the mathematician tells her brother: “What is the sum of the four results you have noted down?”
“243 again,” he says. “What a wonderful coincidence!”
“That’s because both you and Mother used integer values for all four ages,” says his sister, who is 32.
What are the ages of the two children, their uncle, and their grandmother?
#Puzzle 120.2
These are 10 common English words, each consisting of 7 or 8 letters. In each word, however, 4 consecutive letters have been left blank for you to fill.
Your condition: the 4 letters you fill must be the name of a bird. Ten common words, ten 4-letter birds within them.
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 119.1
Hi Kabir,
The table shows who is what, and who is watching which movie on which screen in which hall.
Interestingly, this puzzle came at a time when we (3 in number) are about to go for a film festival (Jagran Film Festival, Dec 5-8) which will have different movies running simultaneously in 3 halls in Siri Fort Auditorium, New Delhi. We haven't yet decided who will watch what (and miss what), but I am making sure that I respond to the puzzle well before this event starts!
— Professor Anshul Kumar, Delhi
Hi,
In this puzzle, POAQ is a rectangle. We have to draw the other diagonal of the rectangle i.e. OQ and we find that it is the radius of the circle. The diagonals of the rectangle are equal. Therefore the length AP is the circle's radius, i.e. 10cm.
— Ajay Ashok, Delhi
Sanjay Gupta adds a fun fact: In this case A is the midpoint of OB. Had it been any other point on OB, AP would still have been 10cm. In other words, Q could be anywhere on this circle, the hypotenuse it subtends on any two right-angled radii would always be 10cm.
Solved both puzzles: Professor Anshul Kumar (Delhi), Ajay Ashok (Delhi), Sanjay Gupta (Delhi), Sabornee Jana (Mumbai), Anil Khanna (Ghaziabad), Yadvendra Somra (Sonipat), YK Munjal (Delhi), Shri Ram Aggarwal (Delhi), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), Shishir Gupta (Indore)
Solved #Puzzle 119.2: Dr Sunita Gupta (Delhi)