The Impossible Problem

In 1969, Hans Freudenthal posed a puzzle that Martin Gardner would later call “The Impossible Problem”. Below is a 2000 version due to Erich Friedman.

I have secretly chosen two nonzero digits and have separately told their sum to Sam and their product to Pam, both of whom are honest and logical.

Pam says, “I don’t know the numbers”.
Sam says, “I don’t know the numbers”.
Pam says, “I don’t know the numbers”.
Sam says, “I don’t know the numbers”.
Pam says, “I don’t know the numbers”.
Sam says, “I don’t know the numbers”.
Pam says, “I don’t know the numbers”.
Sam says, “I don’t know the numbers”.
Pam says, “I know the numbers”.
Sam says, “I know the numbers”.

What are the numbers?

This beautiful problem may at first seem impossible, as you know neither the sum nor the product of the numbers, but the attached animation illustrates my solution.

Animated solution of "The Impossible Problem". Matrix rows & columns are products and sums of nonzero digit pairs; filled squares indicate products & sums shared by pairs and not yet excluded by Pam or Sam

Animated solution of “The Impossible Problem”. Matrix rows & columns are products and sums of nonzero digit pairs; filled squares indicate products & sums shared by pairs and not yet excluded by Pam or Sam

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Electronic Kilogram

The kilogram is the only metric unit still defined by an artifact. The International Prototype Kilogram, IPK or “Le Grand K”, is a golf-ball-sized platinum-iridium cylinder in a vault outside Paris. This year I expect the General Conference on Weights and Measures to replace the IKP by an electronic realization that balances gravitational and electrical power.

The Kelvin or Ampere balance suspends a horizontal wire loop of mass m, length \ell, and current I, by a radial magnetic field B. Integrate the magnetic force \overrightharpoon{F}=q\overrightharpoon{v}\times\overrightharpoon{B} around the loop to find the force balance

m g = F = \left| \oint d \overrightharpoon{F} \right| = \left| \oint dq \,\overrightharpoon{v} \times \overrightharpoon{B}\right| = \left| \oint I d\overrightharpoon{\ell} \times \overrightharpoon{B}\right| =I \ell B

and solve for m. Unfortunately, \ell and B are difficult to measure accurately.

In 1975, Bryan Kibble proposed the calibration step of moving the current-less wire loop vertically at speed v. Integrate the force per charge \overrightharpoon{F}/q=\overrightharpoon{v}\times\overrightharpoon{B}  around the loop to find the induced voltage

V = \oint \overrightharpoon{E} \cdot d\overrightharpoon{\ell} = \oint \frac{\overrightharpoon{F}}{q}\cdot d\overrightharpoon{\ell} = \oint \overrightharpoon{v} \times \overrightharpoon{B} \cdot d\overrightharpoon{\ell} = v B \ell.

Eliminate \ell and B from the force and voltage expressions to find the virtual power

P = V I = v B L I = m g v

in Watts, and again solve for m. Accurately measure voltage V by comparing to the superconducting Josephson-effect voltage

V =\frac{n_J f}{K_J},

where K_J = 2 e / h = 0.48~\text{THz} / \text{mV} is the Josephson constantn_J is the number of Josephson junctions, and f is their microwave frequency. Convert current I = V_R / R to voltage and resistance by Ohm’s law. Accurately measure resistance R by comparing to the quantum Hall-effect resistance

R =\frac{R_K}{n_L},

where R_K = h /e^2 = 26~\text{k}\Omega is the von Klitzing constant, and n_L is the number of filled Landau levels. Accurately measure velocity v and acceleration g using interferometers.

Hence the mass

m=\frac{VI}{gv}=VV_R\frac{1}{R}\frac{1}{gv}=n_J f\left(\frac{h}{2e}\right) n_J f_R\left(\frac{h}{2e}\right) n_L\left(\frac{e^2}{h}\right)\frac{1}{gv}=\frac{n_L n_J^2 f f_R h}{4gv}\propto h,

where h=0.66~\text{zJ} / \text{THz} is the Planck constant. The Kibble or Watt balance thus defines mass in terms of the rate of change of a photon’s energy with its frequency.

The NIST-4 Kibble balance has measured Planck's constant to 13 parts per billion and is thus accurate enough to help redefine the kilogram

The NIST-4 Kibble balance has measured the Planck constant to 13 parts per billion and is thus accurate enough to help redefine the kilogram. Credit: Jennifer Lauren Lee.

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Taylor Bowl

On Wednesday, September 13, 1989, I met with newly elected Physics Club officers Tom Taczak ’91, Dennis Kuhl ’90, Doug Halverson ’91, and Karen McEwen ’90 in Westminister House. I wrote in my diary, “first phys club meeting w. officers goes well”. That year we invented Taylor Bowl, an annual bowling competition between the Physics and Math clubs, both denizens of Taylor Hall, at the bowling lanes in Lowry Student Center. We intentionally chose an activity that either club could do, but that neither club could do well. The annual event was a great success for both clubs for nearly 30 years, but it ends with the demolition of Scot Lanes this month.

Physics at Taylor Bowl Montage

Physics at Taylor Bowl Montage

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Newton’s Can(n)on

One of my favorite illustrations is the cannon thought experiment from volume three of Isaac Newton‘s Principia Mathematica. Johannes Kepler argued that planets orbit elliptically with Sol at one focus. Galileo Galilei argued that terrestrial bodies fall parabolically in space and time. Living in the next generation and standing on their shoulders, Newton realized that Kepler’s ellipses and Galileo’s parabolas were extremes of the same continuum, the Newtonian synthesis, which he dramatized by imagining a cannon on a tall mountain shooting cannon balls at increasing horizontal speeds: a falling apple orbits Earth (but collides with its surface); the orbiting Luna falls toward Earth (but its tangential velocity prevents a collision).

In Newton's famous thought experiment, subsuming both Galileo and Kepler, cannonballs shot at ever increasing horizontal speed eventually fall around Earth

In Newton’s famous thought experiment, subsuming both Galileo and Kepler,
cannonballs shot at ever increasing horizontal speed eventually fall around Earth

Low-resolution photograph of page 6 volume 3 of Newton's Principia, as it appears in the Voyager interstellar record now en route to the stars

Low-resolution photograph of page 6 volume 3 of Newton’s Principia,
as it appears in the Voyager interstellar records now en route to the stars

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ein Stein

I’ve been fascinated by aperiodic tilings of the plane since Martin Gardner first wrote about them in Scientific American. In the 1960s, Robert Berger discovered a set of 20 426 prototiles or tile-types that can tile the plane but only with no translational periodicity — a wonderful mix of the expected and the surprising, a kind of visual music.

Over the years, the number of required prototiles has been greatly reduced. In the 1970s, Roger Penrose discovered a set of just two concave aperiodic prototiles. Robert Ammann then dissected these to discover a set of three convex aperiodic prototiles. Can a single prototile, one tile or stone, literally ein Stein in German, force a nonperiodic tiling? Despite several near misses and potential applications to quasicrystals, the existence of an ein Stein remains a fascinating unsolved problem.

Three convex Ammann tiles force a non periodic tiling of the plane.

Three convex Ammann tiles force a nonperiodic tiling of the plane.

Two concave Penrose tiles force a nonperiodic tiling of the plane.

Two concave Penrose tiles force a nonperiodic tiling of the plane.

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Gossamer Flight

As a kid, I devoured the pages of Popular Science magazine and was fascinated by the quest for human-powered flight: Was a flying bicycle possible?

In the mid 1970s, I read that aerospace engineer Paul MacCready had assembled a team to build a large, lightweight, human-powered aircraft that could be rapidly repaired and redesigned. In 1977, after multiple iterations, cyclist Bryan Allen flew MacCready’s Gossamer Condor around a one-mile figure-eight course to win the first Kremer prize. Two years later, Allen flew MacCready’s improved Gossamer Albatross 22 miles across the English Channel to win the second Kremer prize.

Made with a carbon fiber frame and polystyrene ribs covered with transparent plastic film, each Gossamer aircraft had a long tapering wing behind a large horizontal stabilizer. Weighing less than the pilot-engine, the required power was only about 0.3 kW (or 0.4 hp). Currently, an outstanding Kremer prize is to fly a 26 mile marathon course in under an hour.

Bryan Allen powers and pilots Paul MacCready's Gossamer Albatross across the English Channel in 1979.

Bryan Allen powers and pilots Paul MacCready’s Gossamer Albatross across the English Channel in 1979.

Allen flies MacCready’s Gossamer Albatross II in NASA tests in 1980.

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The Cupola

In the sky is a castle, built in free fall, brick-by-brick, where the sun rises and sets every ninety minutes. The castle derives its energy from sunlight and recycles its water. Sealed against a vacuum, its inhabitants float and glide through its passageways and gaze down at Earth through its expansive cupola.

In an earlier age, the castle would be the magic of legend, but in ours, it’s the International Space Station. Assembled in low Earth orbit, its unique microgravity laboratories are powered by giant solar electric panels that rotate like windmills to track the sun. Arguably the most complex engineering project ever accomplished, the ISS is a model for international cooperation, where former cold-war enemies live and work together.

Astronaut Tracy Caldwell Dyson gazes down at Earth from the cupola onboard the International Space Station

Astronaut Tracy Caldwell Dyson gazes down at Earth from the International Space Station’s cupola

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The Falls

1930s businessman Edgar Kaufmann Sr. and his family lived in Pittsburgh Pennsylvania. Kaufmann owned a rural retreat outside the city and wanted a weekend home there. He assumed his 67-year-old architect would design the home with a good view of the Bear Run waterfall.

Instead, the architect designed the home on the waterfall.

Frank Lloyd Wright’s Fallingwater masterpiece is a 3.5 hour drive from Wooster and makes a wonderful day trip. In 2013 I thoroughly enjoyed an in-depth guided tour of this iconic residence. I look forward to returning some day.

Frank Lloyd Wright's Fallingwater (CC0 1.0 Public Domain)

Frank Lloyd Wright’s Fallingwater in rural Pennsylvania is not far from Wooster. CC0 1.0 Public Domain.

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Variable stars with the Wooster observatory (Jr IS guest blog by Nate Moore)

 

The night sky is full of wonder and splendor. Stars, many more than one can count by themselves, and what a great expanse it truly is, reaching beyond our visible universe. In the vast nothingness, there are things that we can still learn through observation. The first step to learning though is by making sure we have the equipment to do so. My junior independent study consisted of using the Wooster Observatory, to look at the apparent brightness of one of these stars. Despite my [wrong] preconceptions, the stars do in fact change their brightness. Even more surprising, at least to me, was the concept that the stars who are part of constellations also had this happen to them.

I studied, by using a scientific camera and the observatory, the very specific star, Mekbuda which is part of the constellation Gemini. It has a period of about 10 days, which is pretty short compared to others, and changes its apparent brightness by about 0.5 magnitudes, which is a pretty significant change in apparent brightness. I measured and then plotted this data. I must admit, that when I first started this project I had deep concerns that I would not be able to go out to the observatory to collect data as Ohio’s weather does not have the tendency of being friendly towards astronomical research. However, in five weeks, the heavens did clear, permitting for ten days of data collection; five of which were actually used to take data on Mekbuda, three on learning how to do things properly, and two missed by accident.

With such a small amount of data, it seemed unlikely to me that I would be able to get results that had agreement with the currently known data on Mekbuda. However, with stout labor and good science came results which agreed well with the current data (this may be a consequence of having seen three shooting stars in one night though). All it took was a scientific camera, a school’s telescope, a laptop, and the willingness to do science.

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Storing Memory in Light (Jr IS guest blog by Avi Vajpeyi)

When we say that two particles are quantumly entangled, we mean that the particles cannot be looked at independently even when separated by great distances. This means that if we measure one particle, we will automatically get the measurements of the other.

This concept is spooky, and can be useful in studying the nature of the universe. We can also use this as a form of instant communication. If we take a quantum-entangled photon pair and separate them, altering one would alter the other automatically. Hence we would be able to send information automatically.

To be able to use quantum entangled particles to help us communicate over large distances instantaneously,  we would need to first transfer these particles over large distances. The problem with doing so currently is that on the way, sometimes entanglement is lost!

 

FIG1: Quantum Entanglement of Two Particles: A cartoon depicting the spooky entangled relationship between two particles.

FIG2: Entanglement getting lost: The entanglement of photons can be lost over large distances.

To fix this issue, scientists have proposed the use of quantum repeaters. These are devices which accept an entangled photon, save its entanglement in a cluster of atoms, and then emit another photon which is now the new quantum-entangled photon. In a way, these are kind of like pit-stops for the quantum-entangled photons as they travel.

The way the entanglement of a photon can be saved onto a cluster of atoms is tricky: it can involve the photon mapping its phase (the current point in the wave of a light wave) onto atoms. The phase that is mapped onto the atom is called the geometric phase. This phase is very interesting as it stores information about the previous polarisation states — the different orientations of how the electric field in light vibrates — that the light has been through.

This is very peculiar and amazing! By saying that the light carries  geometric phase which saves information about the previous polarisation states the light has been through, we are saying that light knows about its history. For example, when you travel from Wooster to NYC and back, and when you travel from Wooster to Florida and back, you can distinguish both journeys. This is because they were both different paths, although your final start and end points were the same (Wooster). The geometric phase in light shows that even light stores the information needed to distinguish the different previous polarization paths. Additionally, with the geometric phase, we can determine if light has been through a path of polarisation states or not. For example we can distinguish between a person who has travelled from Wooster to NYC and back, and another person who has stayed in Wooster. All this information about the photon’s polarisation path history is given by the geometric phase!

 

FIG3: Different journeys give different geometrical phases: The red dot represents the starting and ending point of the path (Wooster), and its polarisation state can be seen on the right. The path in figure (a) has a geometrical phase of 8.19 and the path in figure (b) has a geometrical phase of 6.26. These separate paths are analogous to the journies to NYC and Florida from Wooster. The video below shows how different points along the path have different polarisation states (similar to different gas stations along the way to NYC / Florida).

For my Junior Independent Study, I studied how light can retain memory using geometrical phase. A previous Wooster Student, Drew King-Smith, began this study using a model with simple constraints. Working with Dr. Cody Leary, I expanded upon this model by considering factors which would make the model more applicable to our equipment in the laboratory. Through the model we were able to produce predictions for numerous interferograms (interference patterns of light), which tell us about the geometrical phase. These predictions can guide future experimental work in the verification of these interferograms. By doing so we would learn more about the physics of quantum memory!

FIG4: Predicted interferograms: These are three interferograms that we predict are created with various parameters.

 

 

 

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