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Problematics | Drone up and down

Oct 28, 2024 02:20 PM IST

A drone moves back and forth between two walkers; what is the distance it travels before the two meet?

You may have come across this one before in some form or the other, because it is a staple in many puzzle books. Although we try to be original for the most part in Problematics, this puzzle is so good that it had to make an appearance here sooner or later. And now is as good a time as any.

Representational image.(Pixabay)
Representational image.(Pixabay)

The puzzle involves an object in motion between two moving objects. For the sake of semi-originality, the most we can do is modify it into yet another form, as puzzler after puzzler has done over the decades.

#Puzzle 114.1

Two security officers set out to clear a 10km stretch of ground of any potentially unsafe objects. Starting from opposite ends, each officer combs the ground with their security devices, centimetre by centimeter, metre by metre, while walking towards the other end.

Just in case the officers miss any object or step on something unsafe, a drone above helps them with its electronic eye. At the instant the two officers start out from opposite ends, the drone too takes off from the same spot as one of them. Now this is one very careful officer, who walks at only 1.5km/hr to make sure he has covered every inch of the ground (that’s just a figure of speech, so let’s not let the non-metric unit disturb our puzzle).

As the officer walks, the drone above travels a little faster, doing 2km/hr so that it can scan the ground ahead, and alert either of the two officers of anything potentially unsafe in front of them. All the while, of course, the other officer is still walking from the other end. Remember, the two officers and the drone all started at the same instant.

The second officer is even more careful than the first and walks slower, at just 1km/hr. The drone, meanwhile, travelling faster than either officer, reaches the second officer at one point. Without wasting an instant, it turns around and starts moving in the opposite direction, still at the same 2km/hr.

Meanwhile, the first officer keeps walking at 1.5km/h and meets the drone after a while. The drone turns again and starts travelling towards the second officer, until it meets him again, at which point it will turn back without wasting a moment.

This goes on and on, the drone travelling back and forth between the two officers. Eventually, the two meet and the drone lands.

How far has the drone travelled in the air?

#Puzzle 114.2

This is another one of those puzzles in which some people always tell the truth, some always lie, and some alternate between truth and falsehood. The book I got it from attributes it to Hubert Phillips, a puzzler who was also an authority on two cards.

You have three people in front of you, one of each kind described above, and have no idea which one is always truthful, which one is a liar, and which one alternates between telling the truth and lying. In fact, you don’t even know whether the alternator will speak the truth first and lie the next time, or whether he will go the other way round.

In your hand is a deck of playing cards that may help you find out. You pick out any card at random and show it to them: “What colour is the suit of this card?”

“Red,” says the first person. “Red,” agrees the second. “Black,” says the third.

You pick out a second card and go through the same routine: “And what colour is this?”

“Black,” says the first person. “Red,” says the second. “Red,” agrees the third.

Are the two cards the same colour? If so, are they both red or both black? And if they are different colours, is it possible to work out in which order they were shown?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 113.1

Hi Kabir

Suppose there are n houses, and the brother adds the house numbers from #1 to #k, whereas the sister adds the house numbers from #n to #k+1. The brother's subtotal = k(1 + k)/2 = (k² + k)/2, and the sister’s subtotal = (n – k)(k + 1 + n)/2 = (n² – k² + n – k)/2. Since the two subtotals are equal,

(n² – k² + n – k)/2 = (k² + k)/2

The (k, n) pairs found satisfying the required equation are: (2, 3), (14, 20) and (84, 119). As the number of houses has 3 digits, the right answer is 84, 119. In other words, the two children met outside house number #85 and the total number of houses is 119.

— Professor Anshul Kumar, Delhi

Anshul Kumar found the answers after determining some constraints for the equation. His working is a little too lengthy to be discussed in detail here.

#Puzzle 113.2

113.2

Hello Kabir,

Say the speed of the boat is x, the speed of the river is y, and the distance one way is d. The times taken are d/(x – y) = 7 (upriver) and d/(x + y) = 5 (downriver). Dividing and solving gives x = 6y. Substituting this in the first equation gives d/(5y) = 7, or d/y = 35 hours. This is the time taken by the raft, which travels at the same speed as the river.

— Dr Sunita Gupta, Delhi

Solved both puzzles: Professor Anshul Kumar (Delhi), Dr Sunita Gupta (Delhi), Akshay Bakhai (Mumbai), Yadvendra Somra (Sonipat), Shishir Gupta (Indore), Kanwarjit Singh (Chief Commissioner of Income-Tax, retired)

Solved #Puzzle 113.2: Anil Khanna (Ghaziabad), Ajay Ashok (Delhi), YK Munjal (Delhi)

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

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