Problematics | Walking with a dinosaur
A brachiosaurus and a human walk the entire circumference of the planet together. Their feet obviously travel the same distance, but what about their heads?
Although dinosaurs went extinct millennia before humans evolved, their old fossils and modern simulations have given us a fair idea of how gigantic some of them were compared to the humans of today. The internet is replete with data while some helpful souls have even drawn comparative illustrations. I found one of various dinosaurs, among which the brachiosaurus would have towered over tall modern humans had they shared the planet at the same time. The difference in height, as you will see, is relevant to the following puzzle.

#Puzzle 97.1
"The earth is round," says the brachiosaurus, who has survived the dinosaur extinction and is living a peaceful vegetarian life among carnivorous humans.
“It's not,” insists the human, a flat-earther whose species has not yet reached the age of scientific awakening.
“There's only one way to prove it,” sighs the brachiosaurus, “and that is to take a walk across the earth’s entire surface. I have no idea how long it will take us, though.”
“If we walk long enough, we will eventually reach one end of the flat earth,” says the human. “Just make sure that neither of us falls off.”
And so they set off on a long walk. After a few days, it’s the human who blinks first. “I am tired,” he says. “I guess it’s up to future generation to develop the tools that will establish the earth’s flatness.”
“We can prove its roundness in this very day and age,” the dinosaur teases him. “Let’s keep walking and we will eventually return where we started.”
“I have walked long enough,” says the human, “and so have you.”
“My feet have travelled as far as your feet,” the dinosaur agrees, “but my head has travelled a longer distance than your head.”
This confuses the human, so the brachiosaurus explains: “Your feet and mine have walked the same arc on the surface of the earth. But my head, being at a height of 18m compared to your height of 1.8m, has described the arc of a larger circle than your head has done.”
“No,” says the human, “we walked along a straight line, so it’s been the same distance for both of us, head and foot.”
The brachiosaurus ignores him. “Suppose both of us walked the entire circumference of the earth. I wonder how much farther my head would have travelled than your head. Had I known the circumference of the earth, I could have calculated all that in my head."
Since you know the circumference is 40,000km, can you help the brachiosaurus work it out?
#Puzzle 97.2
In the following two equations, a, b, x and y are all integers.
a² – b² = x³
a³ – b³ = y²
What are the smallest values you can find for a and b?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 96.1

Dear Kabir,
The meeting point of the two slanting lines lies at a distance of 1.2cm from the bottom horizontal line. I have solved it with different methods, using similar triangles and again using coordinate geometry.
Method #1: Triangles AEF and BCD are similar; therefore, h/2 = AE/AC. Again, from similar triangles CEF and CAB, h/3 = CE/CA. Adding these two equations, we get 5h/6 = (AE + CE)/AC = AC/AC = 1; therefore, h = 6/5 = 1.2cm.
Method #2: The straight line connecting B and C is given by the equation (y – 3)/(x – 0) = (0 – 3)/( x₂ – 0), or, x = x₂(3 – y)/3. Similarly, the line between A and D is given by the equation (y – 0)/(x – 0) = (2 – 0)/(x₂ – 0), or, x = y(x₂)/2. These two lines meet at (0, h), so solving the equations will give us the y-value for h. Solving, x₂(3 – y)/3 = y(x₂)/2, we get y = 1.2cm, i.e. h = 1.2cm.
— Sampath Kumar V, Coimbatore
Although only two of Sampath’s methods are being published here, let us acknowledge him for having tried three approaches: two using similar triangles and one using coordinates.
As for the punctuation puzzle of last week, the source was my childhood memory of a joke I had read decades ago. While any reasonable punctuation that conveys the desired meaning has to be counted as a correct answer, here is a reader who has matched the punctuation exactly as described in the joke:
#Puzzle 96.2
Hi Kabir,
The punctuation may be: WHAT! DO YOU THINK I WILL GIVE YOU A HAIRCUT FOR FREE?
— Shishir Gupta, Indore
Solved both puzzles: Sampath Kumar V (Coimbatore), Shishir Gupta (Indore), Akshay Bakhai (Mumbai), Dr Sunita Gupta (Delhi), Yadvendra Somra (Sonipat), Ajay Ashok (Mumbai), Professor Anshul Kumar (Delhi), Harshit Arora (IIT Delhi), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired), YK Munjal (Delhi)
Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com.
