Problematics | Around the world in 79 days

In the 1850s, some three decades before the international Date Line was established, the author and mathematician Lewis Carroll puzzled over a question of time. I quote: “Suppose yourself to start from London at mid-day on Tuesday, and to travel with the sun, thus reaching London again at mid-day on Wednesday. If at the end of every hour you ask the English residents in the place you have reached the name of the day, you must at last reach some place where the answer changes to Wednesday.”

Given that it’s always midday (or noon) at every place the traveller is reaching, Carroll wondered about the point where the day changed from Tuesday to Wednesday. Back in the 1980s, when I described Carroll’s question, a friend of mine dismissed Carroll as being ignorant of the IDL, which actually did not exist when the author posed the question. I think the question, in fact, reveals Carroll’s intelligence and tells us why the IDL was necessary in the first place.

A couple of decades after Carroll posed his question, and still some years before the IDL would be established, the French novelist Jules Verne delved deeper into the same idea in Around the World in 80 Days. Here, however, the protagonist travels in the direction opposite to the sun’s apparent movement; in other words, he goes in the same direction as the Earth’s rotation.

We touched upon this concept in an earlier episode of Problematics, but that was nearly two years ago. We return to it this week, in a more fundamental way.

#Puzzle 138.1

Let us begin with a recap of the novel’s plot. Phileas Fogg of London has made a bet that he can travel round the Earth in 80 days. He does it in 79 days and a little, but thinks he has missed the deadline by a tiny margin. Only on the next day does he realise that he has miscalculated, and collects the bet in the nick of time.

The reason Fogg miscalculated is that he did not account for the time gained in travelling east. He ultimately figures it out: the 80 sunrises he has encountered means that 79 days have elapsed. Although there is no IDL yet, Fogg’s deduction underlines the importance of having such a date line in place.

Verne explains the concept using longitudes and calculating the average time gained every day. Is there any alternative way, without going into longitudes, to show that you will see the sun rise 80 times if you spend 79 days circling the globe in that direction?

#Puzzle 138.2

The 3-4-5 right triangle gives us the only Pythagorean triple in which all three sides have integer values in single digits. There is only other Pythagorean triple in which two of the integer sides consist of a single digit each, and it is easy to find using hit and trial.

Is there any rigid mathematical proof to establish that this solution is unique? I don’t know if such a proof exists, and I am asking this question in the hope that readers will let me know for sure. But do send me the Pythagorean triple with or without the proof.

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 137.1

Hello Kabir,

Here is my answer in brief. Say the price of a can of paint = a, and the price of a tube-light and a fan together = b.

Cans of paint bought = 15; used = 14; remaining = 1

Fans and tube-lights bought = 8 of each; used = 6 of each; damaged = 1 of each; remaining = 1 of each.

Using given data,

9a + 5b = ₹15250

6a + 3b = ₹9600

Solving, we get a = ₹750 and b = ₹1700. The refund is therefore = a + b = ₹2450.

— Dr Sunita Gupta, Delhi

#Puzzle 137.2

Hi Kabir,

The hidden five-letter word is QUERY. I think this solution is unique.

— Shishir Gupta, Indore

Solved both puzzles: Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), Shri Ram Aggarwal (Delhi), Biren Parmar (Bay Area, California), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Anil Khanna (Ghaziabad), Y K Munjal (Delhi), Sampath Kumar V (Coimbatore), Ajay Ashok (Delhi), Sanjay Gupta (Delhi), Professor Anshul Kumar (Delhi), Aishwarya Rajarathinam (Coimbatore), Vinod Mahajan (Delhi), Yadvendra Somra (Sonipat).

Problematics will be back next week. Please send in your answers by Friday night to problematics@hindustantimes.com