Walking on an escalator

Suppose you walk up a escalator that is itself moving up. Do you count more steps or fewer compared to moving up a stationary staircase of the same length? What happens if you walk down while the escalator moves up, or vice versa? How does the number of steps you walk change as you vary your walking speed, or the number of steps you cover with each stride?
Puzzles with escalator steps offer a number of variations. As far as I can remember, they have been used only once in Problematics, and that was in the early days of this column in its current form in HT, close to two-and-a-half years ago. So here's a selection for you, picked up from various published sources and adapted to Problematics-like situations.
#Puzzle 139.1
#139.1A: Professor Pandit walks up a small escalator that is also moving up. Walking one step at a time, she takes 16 steps to reach the top. As an experiment, she walks downstairs on a regular staircase. She then takes the upward-moving escalator again. This time, she walks up two steps at a time, and reaches the top with 12 steps. She takes the staircase down again, but unfortunately cannot continue the experiment because a power failure results in the escalator shutting down.
#139.1B: In a mall not far away, Professor Pandit's colleague Lecturer Lalaji is conducting his own experiment on another escalator, this one moving downward. Like his colleague, Lecturer Lalaji begins by walking in the same direction as the escalator. Walking slowly downward, he reaches the bottom in 50 steps. The next step of his experiment is different, though. From the bottom, he runs up the down-moving escalator. His upward speed now is 5 times his previous downward speed, but he still maintains the discipline of travelling only one step at a time. To accelerate so much without increasing his stride length, indeed, tells us volumes about Lecturer Lalaji's fitness, even though he is not getting any younger. But to come back to more relevant details, he takes 125 steps to reach the top. He would have loved to continue his experiment, but the manager of the mall has observed him on CCTV. 'These professors!' the manager mutters, and switches off the escalator, much to Lecturer Lalaji's disappointment.
#139.1C: In yet another location, two students are conducting an escalator experiment assigned by their teachers Professor Pandit and Lecturer Lalaji. Young Ms Ahmed and her classmate Mr Basu are both walking up an upward escalator. In the time that Mr Basu takes one step, Ms Ahmed takes three. She counts 150 steps, he counts 75. 'How many steps will we see if the escalator's switched off?' the young man wonders. 'That's easily calculated,' his classmate says.
#Puzzle 139.2
The two professors we met on the above escalators, incidentally, are both family persons and have two children each. Each child is either a boy or a girl, and all four are friends, we are told.
MAILBOX: LAST WEEK'S SOLVERS
#Puzzle 138.1
There are several ways that you can avoid relying on longitudes and still explain why any traveller, like Phileas Fogg, will see the sun rise 80 times if they go around the Earth in 79 days, travelling eastward. It was a pleasure going through the various approaches that readers took:
Hi Kabir,
Here is a simple way to explain why a traveller going around the globe in 79 days, travelling eastward, will see the sun rise 80 times. The sun's angular speed with respect to the Earth is 1 rotation per day, westwards, and the traveller's angular speed with respect to the Earth is 1 rotation per 79 days or 1/79 rotations per day, eastwards. Since the two are rotating in opposite directions, their relative angular speed with respect to each other is the sum of their individual speeds = (1 + 1/79) rotations per day = 80/79 rotations per day. Therefore, in the 79 days that the traveller takes for one round, he will see the sun going around 79 x (80/79) = 80 times.
— Professor Anshul Kumar, Delhi
***
One rotation of the Earth involves one sunrise. When a traveller is also moving in the easterly direction, he adds one rotation by coming back to the starting point. Hence one extra sunrise. Anybody travelling east and covering the distance in N days will see N + 1 sunrises.
— Kanwarjit Singh, Chief Commissioner of Income-tax, retired
***
Think of an experiment where you are on a merry-go-round, and your friend is standing outside and watching you. You are running in the same direction as the platform, say clockwise. Now if you move a distance of x and the platform travels a distance of y, the effective distance travelled is (x + y). This gives an analogy of our Problematics puzzle. The traveller and the Earth are rotating in the same direction. So for an outside observer (sun), the total distance covered by the traveller is the sum of the distances by the traveller and the Earth = (1 round by the traveller) + (79 rounds by Earth in 79 days) = 80 rounds in total.
— Sampath Kumar V, Coimbatore
***
Imagine you're on a 12-hour analog clock, and each full revolution is a day. If you move one hour on the clock face each day, you'll complete the circle in 12 days, but you'll have seen the hour hand 13 times. Apply the same idea to Fogg's travel.
— Vinod Mahajan, Delhi
[In Vinod Mahajan's analogy, the Earth is presumed to be stationary while the sun rotates around it. The clock face represents the stationary Earth. The sun (hour hand) and the traveller (you on the clock) must move in opposite directions for this analogy to work. — KF]
#Puzzle 138.2
Hi Kabir,
Other than 3-4-5 (all three sides single digits), the only Pythagoras triple with two integer sides of single digits is 6-8-10. To prove that the solution is unique, we can check all possible combinations with two or all three sides being single digits. This shows that (3, 4, 5) and (6, 8, 10) are unique.
— Shishir Gupta, Indore
To make it clear, a rigorous proof does exist. My sincere thanks to Professor Anshul Kumar and Sampath Kumar V, who have satisfied my curiosity by sending two different proofs to show that the solution is unique. These are, however, too lengthy to present in this column. As Shishir Gupta and a couple of other readers have pointed out, merely examining at the finite number of possibilities is as good a method as any.
Solved both puzzles: Professor Anshul Kumar (Delhi), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Sampath Kumar V (Coimbatore), Vinod Mahajan (Delhi), Shishir Gupta (Indore), Dr Sunita Gupta (Delhi) Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Shri Ram Aggarwal (Delhi), Aishwarya Rajarathinam (Coimbatore), Ajay Ashok (Delhi)

Try Our AI Features

Explore what Daily8 AI can do for you:

Learn with AIFact CheckingText to SpeechAI Summary

Related Articles

The Hindu

6 days ago

• The Hindu

New flowering plant species discovered in Aravali hills landscape near Jaipur

A new flowering plant species with unusual morphological characteristics has been discovered in the rocky and semi-arid landscape of Aravali hills near Jaipur, shedding light on the region's hidden biodiversity. Classified as Portulaca bharat, the plant adds to the list of Indian endemics. With no other populations currently known, Portulaca bharat has been provisionally assessed as 'data deficient' under the International Union for Conservation of Nature Red List guidelines. The plant's narrow endemism and specific habitat requirements make it highly vulnerable to habitat degradation and climate change. The plant was first spotted by researcher Nishant Chauhan, a committee member of the Satpura Biodiversity Conservation Society (SBCS), who noticed a unique Portulaca-like succulent growing from the rock crevice on the dry and rocky hill slopes near the historic Galtaji temple on the outskirts of Jaipur. Mr. Chauhan collected a few live specimens, which were subsequently cultivated under the controlled conditions in Himachal Pradesh's Hamirpur and later in Lucknow to monitor growth patterns, flowering, and morphological stability across environments. 'After a detailed study and comparison with known species, the plant has been confirmed as a new species to science,' Mr. Chauhan told The Hindu on Monday. The species has been described and published in the latest issue of international scientific journal, Phytotaxa, after a thorough examination of the herbarium collections at the Botanical Survey of India (BSI) and other institutions. The genus Portulaca currently comprises about 153 species worldwide, primarily found in tropical and subtropical regions. These succulent plants are known for their toughness, water-storing tissues, and adaptation to extreme environments. In India, 11 species are currently known, including four endemics, mostly distributed in dry and semi-arid habitats. Mr. Chauhan, whose work focuses on the documentation and conservation of threatened plant species, said Portulaca bharat is currently only known from a single location in the Galtaji hills. 'Only 10 individuals of this species were found in the wild. The discovery calls for enhanced field surveys, habitat protection, and ex situ conservation efforts, especially in neglected dry zones like the Aravalis,' he said. The Aravali Range, one of Earth's oldest geological formations, plays host to numerous micro-endemic species and ecologically significant habitats. The discovery of Portulaca bharat from this underexplored and dry deciduous ecosystem has indicated to the hidden biodiversity of the Aravalis and underlined the urgency of protecting it. Portulaca bharat has opposite and slightly conclave leaves and pale-yellow flowers becoming creamish-white towards apex, with the presence of glandular hairs on stamen filaments and thick roots. 'Naming the plant Portulaca bharat after the country serves as a symbolic reminder of India's rich and still-unfolding natural heritage. This species adds to the list of Indian endemics and opens up new avenues for studies in phytogeography, ecology and evolutionary biology,' Mr. Chauhan said. Mr. Chauhan's colleagues in the research team were Amber Srivastava, research associate at BSI, and Sudhanshu Shekhar Dash and Sushil Kumar Singh, both of whom are scientists with BSI in Kolkata and Dehradun, respectively. Mr. Chauhan, hailing from Shimla, earlier pioneered the first ex situ introduction of the critically endangered plant Commiphora wightii in Himachal Pradesh in 2024 and also rediscovered two plant species previously presumed extinct.

NDTV

6 days ago

• NDTV

Astronomers Discover "New Kind Of Climate" On Pluto

Pluto, long reclassified as a dwarf planet, continues to surprise scientists. Astronomers recently detected new features in its high-altitude haze - revealing more than previously known. Once thought to be a frozen, featureless world, Pluto was found to have icy plains and rugged mountains when NASA's New Horizons spacecraft flew past it in 2015. Now, researchers have studied a bluish and multi-layered haze around Pluto, which stretched more than 300 kilometres above the surface. The astronomers noted that the haze is not just a visual oddity, but something that controls the dwarf planet's climate. In the study titled, Evidence of haze control of Pluto's atmospheric heat balance from JWST/MIRI thermal light curves, astronomers revealed that they detected and measured the thermal emission of Pluto's haze. They said that the "observed fluxes indicate that Pluto's haze is composed of Titan-like organic particles as well as hydrocarbon and nitrile ices and demonstrate that the haze largely controls Pluto's atmospheric balance". Pluto's temperatures, climate and general circulation should therefore be substantially affected by the haze across seasons, the study said. "This is unique in the solar system. It's a new kind of climate, let's say," Tanguy Bertrand, who is an astronomer at the Paris Observatory in France, who led the analysis, told Live Science. The findings were published on June 2 in the journal Nature Astronomy. Mysteries of Pluto Pluto's reclassification as a dwarf planet in 2006 sparked debate. Pluto's surface composition, including nitrogen ice and tholins, is still not fully understood. Pluto has a thin atmosphere which freezes and thaws as it orbits the Sun. Pluto's five known moons, including Charon, Nix, Hydra, Kerberos and Styx, pose questions about formation and evolution.

Economic Times

7 days ago

• Economic Times

Pluto-like planet discovered: How big is it and how far is it from Earth?

Researchers believe that 2017 OF201's highly eccentric orbit points to a chaotic origin, possibly caused by a gravitational push from one of the solar system's gas giants in its early days, sending the object as far as the distant Oort Cloud. Pluto-like planet discovered: How big is it and how far is it from Earth? Tired of too many ads? Remove Ads Pluto Like Planet With an Extraordinary Orbit A Decade's Largest Solar System Discovery Tired of too many ads? Remove Ads Possibly Ejected from the Oort Cloud Solar System May Host Hundreds More A Renewed Push for Outer Solar System Exploration FAQs What is 2017 OF201? Tired of too many ads? Remove Ads The object is estimated to be around 700 kilometers (435 miles) in diameter, which is approximately half the size of Pluto. In a significant breakthrough, astronomers have detected a Pluto-like planet situated far beyond the known boundaries of the solar system. Identified as 2017 OF201 , the icy celestial body may represent the largest object discovered in the outer solar system in over a decade, reigniting interest in the unexplored expanses beyond as an extreme trans-Neptunian object (TNO), 2017 OF201 is believed to be around 700 kilometers (435 miles) in diameter—about half the size of Pluto. Though smaller than the famous dwarf planet, its location and orbital characteristics have startled researchers. The object's aphelion, or farthest distance from the Sun, is more than 1,600 times Earth's orbital distance, while its perihelion, the nearest point to the Sun, is roughly 44.5 times that of Earth—comparable to Pluto's orbital reach.'This suggests a highly elliptical and unusual orbit,' noted Dr. Sihao Cheng from the Institute for Advanced Study, Princeton, which led the discovery, as mentioned in a report by planet takes approximately 25,000 years to complete one revolution around the Sun, hinting at a dramatic cosmic using a combination of data from the Dark Energy Camera in Chile and the Canada-France-Hawaii Telescope, the celestial body was captured over seven years and 19 different exposures. Astronomers suggest that if the object's size is verified via radio telescopes, it will earn the title of the largest newly discovered planetary body in the outer solar system since the early 2010s.2017 OF201 joins a short but significant list of massive TNOs, which includes Eris, Pluto, Haumea, Makemake, and Gonggong. It further challenges previous assumptions about the sparsity of large bodies in the far-flung regions of the solar to the researchers, the extreme orbit of 2017 OF201 suggests a tumultuous past. It may have been gravitationally ejected by one of the gas giants early in solar system history, potentially reaching the Oort Cloud—a distant shell believed to host countless icy bodies—before being pulled back inward.'This is a classic case of a planetary body that didn't just form where it now resides. Its path tells the story of encounters, ejections, and returns,' said Dr. Yifan Yang, a collaborator on the study, as mentioned in a report by Kuiper Belt , the donut-shaped region beyond Neptune, was once thought to be relatively empty. However, the discovery of 2017 OF201 raises fresh questions about what lies beyond.'The fact that 2017 OF201 was detectable while spending just 1% of its orbital period near the inner solar system implies that there could be hundreds of similar-sized objects we simply haven't detected yet,' Dr. Cheng discovery provides renewed impetus for studying the Pluto-like planet category and the solar system's unexplored boundaries. It may also guide future missions akin to NASA's New Horizons, which flew past Pluto in 2015 before entering deeper Kuiper Belt emphasize that while humanity has explored deep space with cutting-edge instruments, the very edges of our own cosmic neighborhood remain largely uncharted territory.2017 OF201 is an icy, Pluto-like celestial body classified as an extreme trans-Neptunian object (TNO). It was recently discovered in the far outer regions of the solar system and could be the largest such object found in over a decade.