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# Some $$a^n+b^n$$ Problems in Number Theory

by Mihai Manea

Mathematics Magazine
April, 2006

Subject classification(s): Number Theory | Algebra and Number Theory
Applicable Course(s): 4.3 Number Theory

Several tricks useful in mathematical competitions are discussed.

A pdf copy of the article can be viewed by clicking below. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page.

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B Light and its ideal speed

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1. Jul 21, 2017

stoomart

When I read the speed of light in vacuum is c, does it imply that light doesn't actually travel at this speed in nature? My guess is no, light always travels at c, it's just that in the definition, we're trying to ignore the affects on light from other stuff like the comological constant and the stress-energy tensor.

2. Jul 21, 2017

Staff: Mentor

It does travel at c in nature, in a vacuum. It is slower in a medium:
https://en.wikipedia.org/wiki/Refractive_index
I'm not sure if this is still the record holder, I thought I've read of even a slower experiment, but it's impressing anyway:
https://en.wikipedia.org/wiki/Slow_light

3. Jul 21, 2017

stoomart

4. Jul 21, 2017

Staff: Mentor

Yes, but the few particles in space are so rare, that they hardly would be in the path of a light ray. The question then would be: Does space have a measurable refractive index different from 1? I don't think so.

5. Jul 22, 2017

stoomart

That makes sense, and also brings up the question of whether pressure or other factors in the radiation or matter-dominated eras may have directly affected the speed light traveled, even temporarily like a medium does.

Last edited: Jul 22, 2017
6. Jul 22, 2017

Ibix

Difficult to answer simply, because it depends what you mean by "speed" and what you mean by "speed of light".

In practice the universe was opaque due to the high density. So the speed of light in the sense of "I set up a flash lamp and a mirror, what time does the pulse return" isn't defined, just as the speed of light in a lump of lead isn't really defined.

That said, if (BIG if) you could clear a small volume, you would find that the speed of light was c. This is because spacetime is always locally flat and special relativity applies.

But if (really ginormous if) you cleared a large volume the effects of curvature would become apparent and it's not possible to define velocity in an unambiguous way across the whole volume. In that case the speed of light depends on how you choose to define it.

Possibly the best answer is the local one. The significance of c is that it is a natural conversion factor between units of distance and units of time. It's almost irrelevant to relativity that light travels at c in vacuum.

7. Jul 22, 2017

Staff Emeritus
"We define c to be the speed of light in vacuum but there are no perfect vacua" is the same complaint that there are no frictionless planes or massless pulleys. It's a complaint that leads nowhere, and certainly not to a better understanding.

Oh, and the speed of light in lead is about 1200 km/s.

8. Jul 22, 2017

ZapperZ

Staff Emeritus
Then how do you explain the CODATA exact determination of "c"?

Zz.

9. Jul 22, 2017

stoomart

Not a complaint, but a question has been nagging me for the last few weeks as I've been considering the CMBR, whether any refractive index exists in space that must be accounted for. Per the following, would it be incorrect to say an imperfect vacuum has a refractive index of >1?

The refractive index can be seen as the factor by which the speed and the wavelength of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, and similarly the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and that the frequency (f = v/λ) of the wave is not affected by the refractive index.

The 2006 CODATA recommendations refer to the BIPM for constants with exact values by definition:

Section II touches on special quantities and units, that is, those that have exact values by definition.

Table I lists those quantities whose numerical values are exactly defined. In the International System of Units (SI) (BIPM, 2006), used throughout this paper, the definition of the meter fixes the speed of light in vacuum c

Last edited: Jul 22, 2017
10. Jul 22, 2017

ZapperZ

Staff Emeritus
Yeah, and those values are obtained how? Out of the air (pun intended)?

Every single one of the CODATA values are obtained out of experiments, or derived from them. If we have never had experienced a "true vacuum", as you insisted, then the CODATA values for "c" can never have been obtained in a true vacuum either. Yet, this is how the standards have been defined.

Zz.

11. Jul 22, 2017

stoomart

I want to clarify that the intention of this thread is not to debate the speed of light as a constant, but to understand if light travels slower in the absence of a "true vacuum". It seems the use of "exact" in defining the speed of light is a bit of a stretch (pun intended:), given a measurement uncertainty of 4 parts per billion:

considering the excellent agreement among the results of wavelength measurements on the radiations of lasers locked on a molecular absorption line in the visible or infrared region, with an uncertainty estimated at ± 4 x 10–9 which corresponds to the uncertainty of the realization of the metre,

considering also the concordant measurements of the frequencies of several of these radiations,

recommends the use of the resulting value for the speed of propagation of electromagnetic waves in vacuum c = 299 792 458 metres per second.

Note: The relative uncertainty given here corresponds to three standard deviations in the data considered.

12. Jul 22, 2017

Staff Emeritus
This is getting silly.

Point the first: Do you not agree that you cannot have three independent definitions for the meter, the second and the meter per second (without risking inconsistency)? So what is the problem with picking the second and the meter per second as the units that will be set by definition?

Point the second: The speed of light in intergalactic space differs from c by about a micron per century. Traversing the entire universe, the non-perfect vacuum of space slows light by only a microsecond. Your quibble makes no difference.

Point the third: even if it is impossible to reach perfect vacuum, one can still determine the speed of light there, by measuring the speed of light in media of varying density, plotting the curve, and seeing where it intersects zero.

13. Jul 22, 2017

kimbyd

The normal matter density at the time the CMB was emitted would have been roughly $1\times 10^{-18} kg/m^3$. The density of air at sea level is roughly $1.2 kg/m^3$. The speed of light is changed by a factor of about one part in a million in our atmosphere. The difference is going to be utterly negligible when the CMB is emitted.

14. Jul 22, 2017

stoomart

Thanks V, this is what I was looking for, sorry for the silly quibble.


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## Recursive expectation (part 2)

This post follows the previous recursive expectation post.

### Problem:

There is an equilateral triangle ABC and the bug is in the vertex A. The bug can move only along edges of the triangle. Once it starts going along an edge, it doesn’t change its moving direction until it reaches the next vertex. It starts walking from the vertex A. It takes bug 1 minute to walk between any two vertices. Every time it comes to a new vertex, it makes a random choice about which of other two vertices to go to next. What is the expected time for the bug to return to the starting vertex A?

### Solution 1:

First, completely intuitive solution, without (much) mathematics. It takes one minute for the bug to come to the edge BC. From that edge, each minute there is 1/2 chance that the bug will return to A. Basically we start the experiment that has 1/2 chance of success each time. So, according to the previous post, expected time to get back to A from BC will be 2 minutes, so all together with the first minute, we get the final result – 3 minutes.

### Solution 2:

And now, let’s try to do it pure mathematical way, without much thinking.

Possible solutions for number of minutes (length of trip) to return back to A are 2, 3, 4, … until infinity. If we take one of the possible solutions, let’s say n minutes, how many different paths do we have for this to happen? Basically, we can go from A to B and then go n-2 times between B and C and then the last time, we go back to A, either from B or from C. Another path would be from A to C and then n-2 times between B and C and then the last time, we go back to A. So, for each n, number of possible paths is always 2. Probability for each path of length n is:

$\left ( \frac{1}{2} \right )^n$

So, probability for n-minutes solution to happen is:

$P\left ( n \right ) = \left ( \frac{1}{2} \right )^n \cdot 2 = \left ( \frac{1}{2} \right )^{n-1}$

Now, the solution for the expectation becomes:

$\begin{array}{rcl} E\left ( X \right ) & = & \sum_{n=2}^{\infty }n\cdot\left ( \frac{1}{2}\right )^{n-1} \\ & = & \sum_{n=0}^{\infty }\left [n\cdot\left ( \frac{1}{2}\right )^{n-1} \right ] - 1 \\ & = & \frac{1}{\left ( 1 - \frac{1}{2}\right )^2} - 1 \\ & = & 4 - 1 \\ & = & 3 \end{array}$

which requires some mathematical knowledge, but doesn’t require much of the intuition.

### Solution 3:

This one will actually use recursive expectation (please see the previous post). Expected time from A back to A is equal to expected time from A to edge BC + expected time from edge BC back to A. Expected time from A to BC is 1. Expected time from BC to A (following the logic from the previous post) is:

$\begin{array}{rcl} E\left ( BC\rightarrow A \right ) & = & \frac{1}{2}\cdot1+\frac{1}{2}\left ( 1 + E\left ( BC\rightarrow A \right )\right ) \\ \frac{1}{2} \cdot E\left ( BC\rightarrow A \right ) & = & 1 \\ E\left ( BC\rightarrow A \right ) & = & 2 \end{array}$

So, the overall expectation is:

$\begin{array}{rcl} E\left ( A\rightarrow A \right ) & = & E\left ( A \rightarrow BC \right ) + E\left ( BC\rightarrow A \right ) \\ & = & 1 + 2 \\ & = & 3 \end{array}$

Recursive expectation once again, provides much easier solution than the mathematical one. Additionally, it’s much more intuitive.

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## Encyclopedia > Logistic equation

Article Content

# Logistic map

Redirected from Logistic equation

The logistic map is an archetypical example of how very complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized by the biologist Robert May[?] in 1976. It was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease, containing two causal loops:

• due to reproduction the population will increase at a rate proportional to the current population
• due to starvation, the population will decrease at a rate proportional to the square of the current population. For example, if there are many foxes in one year, they will eat most of the rabbits, and the next generation of foxes will be small since there is insufficient food.

Mathematically this can be written as

xn+1 = r xn (1 - xn),

where:

• xn is a number between zero and one, and represents the population at year n
• x0 is a number between zero and one, and represents the initial population (at year 0)
• r is a positive number, and represents a combined rate for reproduction and starvation.

By varying the parameter r, the following behaviour is observed:

• With r between 0 and 1, the population will eventually die, independent of the initial population.
• With r between 1 and 2, the population will quickly stabilize on a single value; this value depends on r but does not depend on the initial population.
• With r between 2 and 3, the population will also eventually stabilize on a single value, but first oscillates around that value for some time. Again, the final value does not depend on the initial population
• With r between 3 and 1+√6 (approximately 3.45), the population will oscillate between two values forever. These two values are dependent on r but independent of the initial population.
• With r between 3.45 and 3.54 (approximately), the population will oscillate between four values forever; again, this behavior does not depend on the initial population.
• With r slightly bigger than 3.54, the population will oscillate between 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield the same number of oscillations decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669\cdots[/itex]. All of these behaviors do not depend on the initial population.
• At r = 3.57 (approximately) is the onset of chaos. We can no longer see any oscillations. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated of r that appear to show non-chaotic behavior; for instance around 3.82 there is a range of parameters r which show oscillation between three values, and for slightly higher values of r oscillation between 6 values, then 12 etc. There are other ranges which yield oscillation between 5 values etc.; all oscillation periods do occur. These behaviours are again independent of the initial value.
• Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.

A bifurcation diagram summarizes this. The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x.

The bifurcation diagram is a fractal: if you zoom in on the above mentioned value r = 3.82 and focus on one arm of the three, say, the situation nearby looks just like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals.

A GNU Octave script to generate bifurcation diagrams can be found at Logistic_map/Computer simulation.


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2 added 11 characters in body; edited title

# Calculating the local index of intersection of two algebraic curves.

Let $F_1,F_2$ be two polynomials of two variables $(x,y)$ (say complex variables). Suppose that $F_1$ and $F_2$ have no common factors and $F_1(P)=F_2(P)=0$.

What is in practice the quickest way to calculate the index of intersection of the curves $F_1=0$ and $F_2=0$ at $P$? Or, say, what methods one uses to calculate the index?

I know two definitions of the index but they don't look so handy if $C_1$ and $C_2$ are "complicated" at $P$:

1) Take generic small numbers $c_1,c_2\in \mathbb C$ and count the number of intersections of curves $F_1=c_1$, $F_2=c_2$ close to the point $P$.

2) Calculate the dimension of the vector space $O_P/((F_1,F_2)\cdot O_P)$ where $O_P$ denotes the local ring of $\mathbb C^2$ at $P$.

I guess, there should be one more way to make the calculation, by resolving singularities of curves $C_1$ and $C_2$ at $P$...

But how to do this calculation in an effective way?

1

# Calculating local index of intersection of two algebraic curves.

Let $F_1,F_2$ be two polynomials of two variables $(x,y)$ (say complex variables). Suppose that $F_1$ and $F_2$ have no common factors and $F_1(P)=F_2(P)=0$.

What is in practice the quickest way to calculate the index of intersection of curves $F_1=0$ and $F_2=0$ at $P$? Or, say, what methods one uses to calculate the index?

I know two definitions of the index but they don't look so handy if $C_1$ and $C_2$ are "complicated" at $P$:

1) Take generic small numbers $c_1,c_2\in \mathbb C$ and count the number of intersections of curves $F_1=c_1$, $F_2=c_2$ close to point $P$.

2) Calculate the dimension of the vector space $O_P/((F_1,F_2)\cdot O_P)$ where $O_P$ denotes the local ring $\mathbb C^2$ at $P$.

I guess, there should be one more way to make the calculation, by resolving singularities of curves $C_1$ and $C_2$ at $P$...

But how to do this calculation in an effective way?


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+0

# I really need this answer

0
236
1
+90

Today there is a 60% chance of rain, a 30% chance of lightning, and a 15% chance of lightning and rain together.

1) Determine if rain today and lightning today are independent events?

A)The two events are independent because P(lightning) ⋅ P(rain) = 0.15 does not equal P(lightning and rain) = 0.18.

B)The two events are not independent because P(lightning) ⋅ P(rain) = 0.15 does not equal P(lightning and rain) = 0.18.

C)The two events are independent because P(lightning) ⋅ P(rain) = 0.18 does not equal P(lightning and rain) = 0.15

D)The two events are not independent because P(lightning) ⋅ P(rain) = 0.18 does not equal P(lightning and rain) = 0.15

2) Now suppose that today there is a 60% chance of rain, a 30% chance of lightning, and a 10% chance of lightning if it’s raining. What is the chance of both rain and lightning today?

A)3%

B)6%

C)12%

D)18%

Mar 29, 2019

#1
+6046
+1

$$\text{If rain and lightning are independent then }\\ P[\text{rain AND lightning}]=P[\text{rain}]P[\text{lightning}]\\ 0.15 \neq 0.6 \cdot 0.3 = 0.18\\ \text{that's choice C}$$

$$P[\text{rain AND lightning}] = \\ P[\text{lightning |raining}]P[\text{raining}] = \\ (0.1)(0.6) = 0.06 = 6\%\\ \text{Choice B}$$

.
Mar 29, 2019


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Here's the question you clicked on:

## Beatles $\int\limits \frac{dx}{x \sqrt{81x ^{2}-16}}$ one year ago one year ago

• This Question is Closed
1. agentx5

Trig Sub! I know how to do these now! :-D One moment, I'll show you steps... I need to get my lunch from the microwave oven :-3

2. agentx5

First while I go to get this food, 81 = 9^2 and 16 = 4^2

3. agentx5

Trig sub. For this form with the 1/(x*sqrt(b^2*x^2-a^a)).... let $$x = \large\frac{4}{9}\sec\theta$$ $$dx = \large\frac{4}{9}\tan\theta\sec\theta$$ @Beatles watch what happens when you put this in place for "x" and "dx" :-)

4. agentx5

$\int\limits \frac{1}{x\cdot (9^2x^2-4^2)^{1/2}}\cdot dx$ $\int\limits \frac{1}{x\cdot (9^2(\sec\theta)^2-4^2)^{1/2}}\cdot (\tan\theta\sec\theta \ \ d\theta)$

5. agentx5

Take the unit circle identity for sine and cosine, divide both side by cos^2 $\frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}$ Can you see what happens here, @Beatles? Can you write the identity here in terms of tangent-squared now?

6. agentx5

Still there m8? :-D

7. experimentX

lol ... the fastest way ... do opposite.

8. Beatles

can we use the theorem of inverse trigo func. -_-

9. agentx5

$\int\limits \frac{1}{\sec\theta \cdot (9^2(\sec\theta)^2-4^2)^{1/2}}\cdot (\tan\theta\sec\theta \ \ d\theta)$ @experimentX actually faster way is to use the integral tables ;-)

10. agentx5

Err I mean wait that's 4/9 sec theta

11. agentx5

Let me fix that...

12. experimentX

|dw:1342807447734:dw|

13. experimentX

|dw:1342807585448:dw|

14. experimentX

well ... note the pattern .. rest is just some scrupulous manipulation.

15. agentx5

$\int\limits\limits \frac{1}{\frac{4}{9}\sec\theta\cdot (9^2(\frac{4}{9}\sec\theta)^2-4^2)^{1/2}}\cdot (\frac{4}{9}\tan\theta\sec\theta \ \ d\theta)$ $\int\limits\limits \frac{1}{\frac{4}{9}\sec\theta\cdot \sqrt{16\tan^2\theta}}\cdot (\frac{4}{9}\tan\theta\sec\theta \ \ d\theta)$ $\frac{1}{9} \int\limits\limits \frac{9}{4} d\theta$

16. experimentX

still i recommend the trig substitution agentx5 did

17. agentx5

Things cancel out, a lot. I get that on my scratchpad

18. agentx5

|dw:1342807846986:dw|

19. agentx5

SOH-CAH-TOA: SOH : $$\sin \theta = \large \frac{opposite}{hypotenuse}$$ CAH : $$\cos \theta = \large \frac{adjacent}{hypotenuse}$$ TOA : $$\tan \theta = \large \frac{opposite}{adjacent}$$ $$\huge \frac{1}{\cos\theta} = \sec\theta$$ Therefore: $$\huge \theta = sec^{^-1}(\frac{hyp}{adj})=sec^{-1}(\frac{9x}{4})$$ Agreed @TuringTest & @experimentX , and do you follow what I'm doing here so far @Beatles?

20. agentx5

We're going to use this to replace theta after you do that (now simple) integral above

21. experimentX

well ... i call that more intuitive and geometrical approach.

22. experimentX

|dw:1342808370968:dw| the answer would be |dw:1342808423489:dw| sorry ... i forgot chain rule.

23. agentx5

$\huge \frac{1}{9} \int\limits \frac{9}{4} d\theta = \frac{1}{\cancel{9}} \left[ \frac{\cancel{9}\theta}{4} \right] = \frac{\sec^{-1}(\frac{9 x}{4})}{4}+C$

24. agentx5

That should be the answer I think ^_^


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## 14. D.C. Circuits

The Potential Divider Rule

Determining the relative brightness of light bulbs using the potential divider rule

How Internal Resistance affects brightness of light bulbs

## Electromagnetic Induction and its Effects

To explain a phenomenon that happens due to Electromagnetic Induction, we can use an acronym CFILE to structure our answer.

C stands for cutting of flux or changing of flux linkage. This is necessary for electromagnetic induction to happen. When a wire is pulled through a magnetic field perpendicular to it, it is said to be cutting the field lines. When a magnet enters a coil of wires, we can say that the magnetic flux linkage is increasing.

F stands for Faraday's law, which states that the induced e.m.f. in a circuit is directly proportional to the rate of change of flux-linkage or to the rate of cutting of magnetic flux.

I stands for an induced current. However, do note that this is only possible if there is a closed circuit or a path for the current to flow.

L stands for Lenz's law, which states that the direction of the induced e.m.f. is such that it tends to oppose the flux change causing it, and does oppose it if induced current flows

E stands for effect. This is really just stating how an induced current or Lenz's law causes the phenomenon in question.

## Displacement, Velocity and Acceleration of Bouncing Ball using Datalogger

A video tutorial on the use of the Addestation datalogger with its motion sensor to measure the displacement of a bouncing ball and to observe the velocity and acceleration using its differentiation function.

## 17. Alternating Currents

• The root-mean-square value of an alternating current is equivalent to the steady direct current that would dissipate heat at the same rate as the alternating current in a given resistor.
• For a sinusoidal source,
(a) the root mean square value of the current is given by $I_{rms}=\frac{I_o}{\sqrt{2}}$.
(b) the mean or average power < P > absorbed by a resistive load is half the maximum power.
$

=\frac{1}{2}P_o=\frac{1}{2}{I_o}{V_o}=\frac{1}{2}{I_o}^2R =\frac{V_o^2}{2R}$

.
• An a.c. transformer is a device for increasing or decreasing an a.c. voltage. It consists of a primary coil of Np turns and voltage Vp and secondary coil of Ns turns and voltage Vs wrapped around an iron core.
• For an ideal transformer (assuming no energy is lost), the following equation is obeyed
$\frac{N_s}{N_p}=\frac{V_s}{V_p}=\frac{I_p}{I_s}$.
• Power loss in the transmission lines is minimized if the power is transmitted at high voltages (i.e. low currents) since $P_{loss}=I^2R$ where I is the current through the cables and R is the resistance of the cables.
• The equation $P=\frac{V^2}{R}$ is often mistakenly used to suggest that power lost is high when voltage of transmission is high. In fact, V refers to the potential difference across the cables, which often have but a fraction of the overall resistance through which the current passes.

## 18. Quantum Physics

[accordions autoHeight='true']

[accordion title="1. Particle Nature of Light"]

• photon is a quantum of electromagnetic radiation.
• The energy of a photon is given by E=hf, where h is Planck's constant (6.63 $\times$ 10-34 J s) and f is its frequency.

[/accordion]

[accordion title="1.1 Photoelectric Effect"]

• The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency is shone on it.
• The energy of an incident photon is the sum of the maximum kinetic energy $K.E._{max}$ of the emitted electrons from the metal surface and the work function $\Phi$ of the metal. Einstein's photoelectric equation states that

$hf=\Phi +K.E._{max}=hf_o +K.E._{max}$

• where $f_o$ is the threshold frequency or minimum frequency of the electromagnetic radiation below which no electrons are emitted from the metal surface regardless of the intensity of the radiation.
• The work function $\Phi$ of a metal is the minimum energy needed to remove an electron from the metal surface.
• $K.E._{max}$ can be measured by applying a voltage to prevent the emitted electrons from reaching the electrode that collects them. This voltage is known as the stopping voltage $V_s$ and since the charge of an electron is e, the equation can be rewritten as

$hf=\Phi + eV_s$.

[/accordion]

[accordion title="1.2 Line Spectra"]

• An atom is in the ground state when its electron occupies the lowest energy level. When the atom gains energy, its ground state electron makes a transition to a higher energy level. The atom is said to be in an excited state.
• At this excited state, the electron is unstable. It will jump to a lower energy level by emitting a photon whose energy is equal to the energy difference between the two levels. The photon energy is given hf = Ehigher – Elower.
• The emission line spectra are the spectra of light radiated by individual atoms in a hot gas when the electrons in the atoms jump from higher energy levels to lower energy levels. Each spectrum consists of coloured lines on a dark background.
• The absorption line spectra consists of dark lines on a coloured background. When a beam of white light is passed through a cool gas, photons whose energies are equal to the excitation energies of the gas atoms, are absorbed. These photons are re-emitted in all directions, so the intensity of these wavelengths in the transmitted white light beam is reduced.

[/accordion]

[accordion title="2. Wave Nature of Particles"]

• Louis de Broglie postulated that, because photons have wave and particle characteristics, perhaps all forms of matter have both properties. Electron diffraction provides evidence for the wave nature of particles.
• The de Broglie wavelength of a particle is given by $\lambda = \dfrac{h}{p}$ where p is the momentum (mv) of the particle and h is Planck’s constant.

[/accordion]

[accordion title="3. X-ray Spectrum"]

[/accordion]

[accordion title="4. Heisenberg Uncertainty Principle"]

[/accordion]

[accordion title="5. Wave Function and Probability"]

• An electron can be described by a wave function $\Psi$ where the square of the amplitude of the wave function $|{\Psi}|^2$ gives the probability of finding the electron at a point.

[/accordion]

[accordion title="6. Quantum Tunneling"]

• Classically, an electron of energy E approaching a potential barrier, whose height U is greater than E, cannot penetrate the barrier but would simply be reflected and return in the opposite direction.
• However, quantum mechanics predicts that since $|{\Psi}^2|$ is non-zero beyond the barrier, there is a finite chance of this electron tunnelling through the barrier and reaching the other side of the barrier.
• The transmission coefficient T represents the probability with which an approaching electron will penetrate to the other side of the barrier. The transmission coefficient T is given by $T=e^{-2kd}$ where $k=\sqrt{\dfrac{8\pi^2m(U-E)}{h^2}}$

[/accordion]

[/accordions]

## Using a datalogger to measure induced emf

This video tutorial is a guide for next week's practical for CG18/12.

## 20. Nuclear Physics

### The Nucleus

• existence and size demonstrated using the Rutherford $\alpha$-scattering experiment.
• consists of nucleons (protons and neutrons)
• isotopes of an element share the same number of protons but different number of neutrons.

### Nuclear Reactions

• nuclear reactions involve two or more reactants.
• represented using the form: ${^{14}_7N}+{^4_2He}\rightarrow{^{17}_8O}+{^1_1H}$
• for a reaction that releases energy, mass-energy of reactants = mass-energy of products + E,
where $E = mc^2$ and m is the mass defect (difference in mass between the products and reactants).
• binding energy is the energy released when the nucleus is formed from its separate protons and neutrons. The same amount of energy is required to break up a nucleus into its constituent nucleons.
• binding energy per nucleon ($\frac{B.E.}{A}$) is an indication of the stability of a nucleus, where B.E .is binding energy and A is the nucleon number. You need to know how to sketch its variation with nucleon number. (The following video explains the shape of the $\frac{B.E.}{A}$ versus A graph and why it peaks at $^{56}Fe$.
• nuclear fission is the disintegration of a heavy nucleus into two lighter nuclei of comparable mass with the emission of neutrons and release of energy.
e.g. ${^{235}_{92}U}+{^1_0n}\rightarrow{^{236}_{92}U}\rightarrow{^{144}_{56}Ba}+{^{90}_{36}Kr}+2^1_0n+Energy$
• nuclear fusion occurs when two light nuclei combine to form a single more massive nucleus, leading to the release of energy.
e.g. ${^2_1H}+{^3_1H}\rightarrow{^4_2He}+{^1_0n}+Energy$
• The following quantities are always conserved:
• proton number & neutron number
• momentum
• mass-energy

• $\alpha$ particle - helium nucleus
• $\beta$ particle - electron
• $\gamma$ particle - electromagnetic radiation
• $A=-\frac{dN}{dt}=\lambda N$
where A is the rate of disintegration or activity, N is the number of radioactive nuclei and $\lambda$ is the decay constant.
• $x=x_0{e^{-\lambda t}}$
where x could represent the activity, number of undecayed particles or received count rate.
• half-life ($t_{\frac{1}{2}}$) is the average time taken for half the original number of radioactive nuclei to decay.
• From $x=x_0{e^{-\lambda t}}$,
$\frac{x}{x_0}=\frac{1}{2}=e^{-\lambda t_{\frac{1}{2}}}$
$\Rightarrow{-ln2}=-\lambda t_{\frac{1}{2}}$
$\Rightarrow{t_{\frac{1}{2}}}=\frac{ln 2}{\lambda}$
• You may also use ${\frac{x}{x_0}}={\frac{1}{2}}^{\frac{t}{t_{1/2}}}$, as shown in the following video.

## 19. Lasers and Semiconductors

### LASER

• This flow chart shows the requirements expected of students when describing how light is produced using LASER (Light Amplification by Stimulated Emission of Radiation).

### BAND THEORY

• Energy bands are formed in solids where there are multiple atoms, each closely packed with their neighbouring atoms. The outer electrons of each atom interacts with those of neighbouring atoms. Each energy level will then split into a band consisting of many sub-levels. These sub-levels are so close to each other that they coalesce to form an energy band.
• The valence band is the highest band that is completely filled.
• The conduction band is the empty or partially filled band that is just above the valence band.
• The forbidden band is the region where no electron state is allowed.

### EXTRINSIC SEMICONDUCTORS

• In n-type semiconductors, the impurity energy level (or donor level) lies just below the conduction band. Electrons in this energy level can easily reach the conduction band at room temperature. The majority charge carriers are electrons.
• In p-type semiconductors, the impurity energy level (or acceptor level) lies just above the valence band. Electrons from the valence band can easily jump into it leaving behind positive holes. The majority charge carriers are holes.
• The increase in number of charge carriers increases conductivity (and reduces resistivity).


text:
# Circle A has a radius of 3  and a center of (5 ,4 ). Circle B has a radius of 4  and a center of (7 ,2 ). If circle B is translated by <3 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 6, 2017

The circles overlap

#### Explanation:

Let the center of the circle $A$ be ${O}_{A} = \left(5 , 4\right)$

and the center of the circle $B$ be ${O}_{B} = \left(7 , 2\right)$

The center after translation is

${O}_{B} ' = \left(7 , 2\right) + < 3 , 2 > = \left(10 , 4\right)$

The distance between the centers is

${O}_{A} {O}_{B} ' = \sqrt{{\left(10 - 5\right)}^{2} + {\left(4 - 4\right)}^{2}}$

$= 5$

The sum of the radii is

${r}_{A} + {r}_{B} = 3 + 4 = 7$

As ,

${O}_{A} {O}_{B} ' < \left({r}_{A} + {r}_{B}\right)$

The circles overlap

graph{((x-5)^2+(y-4)^2-9)((x-7)^2+(y-2)^2-16)((x-10)^2+(y-4)^2-16)=0 [-6, 16.5, -2.185, 9.065]}


text:
# c++ Checking if a string is an isogram

So I'm starting out and doing some code challenges in between book and tutorial learning and was hoping to get some general feedback on code readability and use of comments.

I know it's sometimes hard to understand your own code after you wrote it so my question is - how easy is it to understand this one based on comments?

Do you think I'm missing anything crucial? Any general feedback welcome, thanks.

/*
https://edabit.com/challenge/SEzYX5qtTR4WcZeR3
An isogram is a word that has no repeating letters, consecutive or
nonconsecutive.Create a function that takes a string and returns
either true or false depending on whether or not it's an "isogram".
Examples
isIsogram("Algorism") ➞ true
// Not case sensitive.
isIsogram("Consecutive") ➞ false
*/

#include <iostream>
#include <ctype.h> // to use the toupper function which takes a letter
character and returns the upper case ASCII integer
using namespace std;

int main()
{
string input_string;
string output_string;
bool isogram_check = true; // by default the program assumes that    everything is an isogram and then tries to find repeating letters to assert that it's not an isogram

cout << "Enter a string." << endl;
cin >> input_string;

for (int i = 0; i < input_string.length(); i++) // this loop takes the input string and converts it to all upper case letters.
{
output_string = output_string + char(toupper(input_string[i]));
}

// cout << output_string << endl;
// the above line is to test the toupper output

for (int i = 0; i < output_string.length(); i++)       // the first loop decides WHAT to compare to something else, the second loops decides what that SOMETHING ELSE is.
{
for (int o = i+1; o < output_string.length(); o++) // this loop takes the character at position i and compares it to the character at i+1 then i+2, etc. by iterating o.
{                                                  // in the above comment, i+1 = o since we iterated i once right at the beginning.
if (output_string[i] == output_string[o])      // when the loop finds two of the same character within the string it brakes out of both loops
{
isogram_check = false;
break;
}
}

if (isogram_check == false)
{
break;
}
}

if (isogram_check == false)
{
cout << input_string << " is not an isogram.\n";
}
else
{
cout << input_string << " is an isogram.\n";
}
}

Avoid using namespace std.

Keep your lines shorter. Having to constantly scroll back and forth to read lines is unpleasant. Most of your comments should be on one (or more) lines right before the statement being commented on. Also, avoid obvious things in comments (like "resets i to use in the next loop.").

Declare variables as close as possible to their first use.

Since isogram_check is a bool, you can test it with !isogram_check instead of isogram_check == false.

With a slight change in algorithm (searching characters before the ith one), your two while loops can be combined into one. Although there are other ways to do this that could be better (for some definition of "better"), your implementation is a quick first pass that works well for short strings (but less well as the strings get longer).

• Thanks. I am intentionally using the namespace declaration since I know I won't be using other namespaces. Should I avoid it even knowing the issue that can come up? What would I use to combine the 2 loops? My idea is that since I need to iterate both the WHAT I'm comparing and the what I'm comparing TO that I need 2 loops. Jan 5, 2020 at 0:03
• @ChirilRussu you never know which namespace you need in the future, so it's best to avoid that. One of my employers' projects uses namespace std everywhere and causes a lot of name clashes problems that we have to do some workaround to avoid to much code change Jan 5, 2020 at 11:37
• @ChirilRussu - Quoting from the zen of python, "Namespaces are one honking great idea — let's do more of those!" The same concept very much applies to C++. The key intent of namespaces is to avoid collisions. It's best to avoid "using namespace <namespace_name>", ever. Jan 5, 2020 at 15:35

I am not an expert in but your code seems verbose. You can try to make it more explicit for a person who knows better better. Plus the algorithm is not that hard to digest so writing some built-in constructions will not overcome the ability to understand the whole flow.

Unless your task is not bound by memory you can use std::set to do the counting of unique characters for you and squeeze out some lines of code.

#include <iostream>
#include <algorithm>
#include <unordered_set>
#include <ctype.h>

int main()
{
std::string input_string;
std::cout << "Enter a string." << std::endl;
std::cin >> input_string;

std::unordered_set<char> chars;

std::cout << input_string << (std::all_of(
input_string.begin(),
input_string.end(),
[&chars](const char c) {
auto upper = toupper(static_cast<unsigned char>(c));
//                                 ^  ^  ^
//                         avoid undefined behaviour
return chars.insert(upper).second;
}
) ?
" is an isogram.\n" :
" is not an isogram.\n");

}

I see you're probably trying to practice for coding interviews or just exercising your problem-solving skills, either way, you should think about using the stl of the given language.

What I mean is not using sorting functions when you are asked to do sorting challenge, but rather use some built-in algorithms as an intermediary step before achieving what your goal is.

• Thanks, I'm just learning that's why I don't know the best ways to do things. For example I don't know how transform works, never used .begin or .end. Don't know how set works, etc. Now that you wrote them for me I can just look it up and figure it out, but it's sort of I don't know what I don't know type of situation while I'm still learning. Jan 5, 2020 at 0:19
• @ChirilRussu, well set is a data structure that only keeps unique elements, so it maintains uniqueness for you. So now you probably see that if set.size() is equal to input_string.size() this means that all chars from input_string got into the set => they are all unique Jan 5, 2020 at 0:26
• @ChirilRussu .begin and .end are methods that return iterator to a certain point of the caller - in this case the set. You can think of them as start and end indexes until you feel comfortable to dig down into en.wikipedia.org/wiki/Iterator_pattern Jan 5, 2020 at 0:28
• @ChirilRussu, transform is just a function that takes start and end iterator (you can think of index) and applies some function to each element that this iterator is traversing in this case its the string Jan 5, 2020 at 0:30

You have two while loops that have a counter i that is incremented. You should really use a for loop.

Your algorithm has quadratic complexity. Can you come up with something that is linear? To be fair, I can immediately only think of a linear solution that has a way higher proportionality constant, so it will be faster only for very long strings. Still.... worth exploring.

• Thanks, I changed both while loops to for loops and got rid of the global i. Jan 5, 2020 at 0:05
• so !isogram_check works, just to clarify my understanding that is the same as isogram_check != initialized value ? Jan 5, 2020 at 0:12
• right, something==false is the same as !something Jan 5, 2020 at 0:15

• code in a function
• limit source code witdth to to 80-100 columns
• code is overly complex for the task, hard to digest it

Try to use the available algorithms and data structures. Those are reference links, but many good articles and books have been written.

One good way to solve many problems related to "frequency of occurrence" is to produce a "count of unique occurrence" or "frequency". Arguably the canonical tool for this is std:unordered_map, which uses a hashtable internally. Often it is combined with std::partial_sort_copy to find the "10 most frequent" or similar. We don't need sort_copy here because we can return false on first repeat. There is an suggested code for the is_isogram() function using std::unordered_map with some static calls from main() below.

But the problem here is even simpler, because we don't actually need to know how many repeats there are I have provide a second function is_isogram_set() which uses an std::unordered_set. I ran some quick benchmarks and the std::unordered_set is about 5-10% faster than the std::unordered_map.

If we can assume "ASCII only" text, and ignore nonalpha completely, then all uppercase'd characters will be between 'A' - 'Z' and we could sensibly use something like a std::array<bool> for which I have provided a third sample is_isogram_array(). This is signignificantly (~10-15x !) faster than the other 2 options.

Any of these should be easily understood by anyone with C++ experience, rather than unpicking "several custom raw loops". Using the standard containers allows us to focus on clean efficient code, rather than "loop correctness". So we were easily able to find significant speed/feature trade-offs.

EDIT: Based on a suggestion from @LaurentLARIZZA I added an alternative version of the std::array variant which uses the STL algorithm std::all_of. I named it is_isogram_array_algo(). I doesn't fundamentally change anything, but it does allow us to avoid the ranged for loop and to make the code slightly more self documenting. ie we can name the lambda "unseen" and write "all of unseen", which arguably makes it slightly clearer what is going on. Code is slightly longer, but has identical performance.

#include <iostream>
#include <string>
#include <cctype>
#include <unordered_map>
#include <unordered_set>
#include <array>

bool is_isogram(const std::string& str) {
auto freqs = std::unordered_map<char, int>{};
for (auto&& c: str) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
int& count = freqs[std::toupper(static_cast<unsigned char>(uc))];
if (count > 0) return false;
count++;
}
}
return true;
}

bool is_isogram_set(const std::string& str) {
auto freqs = std::unordered_set<char>{};
for (auto&& c: str) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
auto [iter, was_new] = freqs.insert(std::toupper(uc));
if (!was_new) return false;
}
}
return true;
}

bool is_isogram_array(const std::string& str) {
auto freqs = std::array<bool, 26>{false};
for (auto&& c: str) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
int idx = std::toupper(uc) - 'A';
if (freqs[idx]) return false;
freqs[idx] = true;
}
}
return true;
}

bool is_isogram_array_algo(const std::string& str) {
auto freqs = std::array<bool, 26>{false};

auto unseen = [&freqs](char c) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
int idx = std::toupper(uc) - 'A';
if (freqs[idx]) return false;
freqs[idx] = true;
}
return true;
};

return std::all_of(str.begin(), str.end(), unseen);
}

int main() {

std::cout << std::boolalpha
<< "Algorism=" << is_isogram("Algorism") << '\n'
<< "Consecutive=" << is_isogram("Consecutive") << '\n';
return 0;
}

• I think it would be better to use auto c : str here. For why I wouldn't use auto&& is here, and copying chars is cheap.
– Juho
Jan 5, 2020 at 16:15
• I don't disagree. In fact i had it that way and changed last minute before posting. Trying to make for (auto&& e: ..) a habit because "it always works": quuxplusone.github.io/blog/2018/12/15/autorefref-always-works Jan 5, 2020 at 16:17
• Thanks for the link, interesting point as well!
– Juho
Jan 5, 2020 at 16:26
• Just did a test for a simplified (to get rid of map and iostream boilerplate code) example on godbolt: godbolt.org/z/2R3YdU I copy pasted the ASM for auto and for auto&& and the diff is precisely nil. I am not saying it will always be, in fact this example is "too simple" as it gets SIMD vector optimised etc etc. But in general could we assume that auto&& will not gen "worse" code? I agree it looks "fishy" if you're not used to it though... Jan 5, 2020 at 16:34
• Would an unordered_multiset also be a fitting data structure to represent the concept of a "histogram"? I don't know C++, but that's what I would use in a different language (codereview.stackexchange.com/a/233584/1581), even going so far as to implement my own (stackoverflow.com/a/57644639/2988). Jan 5, 2020 at 16:42

Building on Oliver Schonrock's answer, here is an equivalent version of the code leveraging the std::all_of algorithm. Note that I did not try to change the semantics of given in Oliver's answer, as all three structures could be made much shorter by always counting, and using bitwise operations.

#include <iostream>
#include <string>
#include <cctype>
#include <unordered_map>
#include <unordered_set>
#include <array>

struct IsFirstOccurrence_UM {
bool operator()(char c) {
int& count = freqs[std::toupper(static_cast<unsigned char>(uc))];
if (count > 0) {
return false;
} else {
count++;
return true;
}
}
private:
std::unordered_map<char, int> freqs;
};

struct IsFirstOccurrence_Set {
bool operator()(char c) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
auto [iter, was_new] = freqs.insert(std::toupper(uc));
return was_new;
} else {
return true; // Note that non-alpha characters are always considered first-occurrence
}
}
private:
std::unordered_set<char> freqs;
};

struct IsFirstOccurrence_Array {
bool operator()(char c) {
if (auto uc = static_cast<unsigned char>(c); std::isalpha(uc)) {
int idx = std::toupper(uc) - 'A';
if (freqs[idx])  {
return false;
} else {
freqs[idx] = true;
return true;
}
} else {
return true; // Note that non-alpha characters are always considered first-occurrence
}
}
private:
std::array<bool, 26> freqs{false};
};

template<typename IsFirstOccurrencePedicate>
bool is_isogram(const std::string& str, IsFirstOccurrencePedicate is_first_occurrence) {
return std::all_of(str.begin(), str.end(), std::move(is_first_occurrence));
}

int main() {

std::cout << std::boolalpha
<< "Algorism=" << is_isogram("Algorism", IsFirstOccurrence_UM()) << '\n'
<< "Consecutive=" << is_isogram("Consecutive", IsFirstOccurrence_UM()) << '\n';
return 0;
}

• Yeah, nice. Demonstrates something slightly different. I also added a version above using std::all_of using a lambda, so without the templated reuse. I viewed the 3 options as mutually exclusive alternatives, and you would keep only one. I think we are beginning to overthink this trivial problem ;-) Jan 6, 2020 at 13:56
• Yes we are. That's why I avoided fleshing out the IsFirstOccurrencePredicate concept :) Jan 6, 2020 at 14:01

As an alternative to the "efficiency" focused options in my other answer, here is an option which is just very terse. Given we can't have words > 26 characters which are isograms, this might be just enough?

#include <algorithm>
#include <unordered_set>
#include <cctype>

bool is_isogram(std::string s) {
std::transform(s.begin(), s.end(), s.begin(), [](unsigned char c) { return std::toupper(c); });
return std::unordered_set<char>(s.begin(), s.end()).size() == s.size();
}

It's closer (!) to what one might write in a language like python.

def is_isogram(s):
return len(s) == len(set(s.upper()))


text:
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# The anatomy of social dynamics in escape rooms

## Abstract

From sport and science production to everyday life, higher-level pursuits demand collaboration. Despite an increase in the number of data-driven studies on human behavior, the social dynamics of collaborative problem solving are still largely unexplored with network science and other computational and quantitative tools. Here we introduce escape rooms as a non-interventional and minimally biased social laboratory, which allows us to capture at a high resolution real-time communications in small project teams. Our analysis portrays a nuanced picture of different dimensions of social dynamics. We reveal how socio-demographic characteristics impact problem solving and the importance of prior relationships for enhanced interactions. We extract key conversation rules from motif analysis and discuss turn-usurping gendered behavior, a phenomenon particularly strong in male-dominated teams. We investigate the temporal evolution of signed and group interactions, finding that a minimum level of tense communication might be beneficial for collective problem solving, and revealing differences in the behavior of successful and failed teams. Our work unveils the innovative potential of escape rooms to study teams in their complexity, contributing to a deeper understanding of the micro-dynamics of collaborative team processes.

## Introduction

From the viral spread of rumours to the emergence of large-scale cooperation, human societies produce social dynamics and collective endeavours often hard to understand, characterize and predict. At the heart of this phenomenon is the innate need and ability of humans to collaborate and connect with others1. Social interactions are indeed key to understand information exchange2 and social contagion3,4,5,6. In recent years, advances in technologies have allowed us to track at unprecedented fine-grained scale face-to-face communication patterns in a variety of different contexts7,8,9,10,11,12,13, drawing attention to the importance of high-frequency time-resolved social processes14 to understand collective intelligence15 and collaborative problem solving11,16,17,18. In the last decades, network science has proved to be a powerful and flexible framework to understand the complex relational structure of human dynamics19,20, from structural balance theory21,22 to the detection of emergent mesoscale structures such as communities23 and cores24, associated with the coordinated behavior of multiple individuals in human societies.

From sport25 to science26, a crucial dimension of social interactions is encoded in teams. Indeed, whenever a goal needs to be achieved, people naturally join forces to overcome the limitations imposed by the capabilities of single individuals. In science, the importance of teams is increasingly recognised, as collective efforts are becoming more and more important for the production of knowledge, not only in volume but also in impact and attention27. The predominance of team success has been attributed to the increasing need for specialized knowledge from different domains recombined through interdisciplinary collaborations to solve modern day problems28,29,30, highlighting the importance of team composition31. Past investigations about drivers of success in teams have revealed a complex and multifaceted picture. Diverse26 and fresh teams32 may help widen skills and perspectives, though potentially introduce conflict and hinder efficient communication33. Other known determinants of team success include collaborations across multiple institutions29, inter-member familiarity34 and prior shared successes of teams35,36. Going beyond the simple dichotomy between solo and synergistic work, also size was found to be relevant, with larger teams developing science by focusing on more recent trends, in contrast to smaller ones, tending instead to dig deeper into past literature and often producing more innovative recombination of ideas37. Even in apparently non-collaborative performances, such as those of elite athletes in individual disciplines such as tennis or martial arts, the presence of an organized, task-focused supportive team of people has become predominant and key for succeeding. All in all, teams serve as ’petri-dishes’ for social influence and are often considered factories of innovation38,39.

In this work we study team dynamics in the innovative settings of escape rooms. Escape rooms are entertainment facilities that can be described as live-action board games, where a group of players has to collaborate intensely (search for clues, open locks, decipher codes) under time pressure to achieve the goal of the game by exiting the room within a predefined amount of time (typically one hour). Escape rooms are free from the typical frailties of traditional laboratory experiments and field studies. Similarly to experiments, they provide the same controlled environment for all groups under observation, yet minimizing the potential modification of participants’ behavior as a response to being examined by researchers40. Besides, social interactions can be followed and recorded at a high-frequency, allowing us to observe intact, non-manipulated teams in a nuanced, meticulous manner, overcoming the limitations associated with temporally aggregated data41,42 or self-reported questionnaires. Exploiting this innovative setting, we extracted from video records the real-time verbal and nonverbal communication of 40 small problem-solving teams. We analyze their high-resolution social dynamics, including collaborative network evolution and conversation rules guiding communication, exchanges of emotions and group interactions, linking them to successful team performance. Besides, we integrate such information by exploring the wider sociodemographic characteristics of team players, including gender, age and education, and their prior acquaintanceship, and meeting frequency. By capturing high-frequency real-time social interactions, our research highlights the potential of escape rooms to investigate task-performing teams in a minimally biased environment, contributing to advance the new science of teams.

## Results

We study the social interactions of 40 teams collected in two distinct escape rooms. For each team (composed of 4 or 5 individuals, for a total of 171 players, all from Hungary) we extract high-resolution temporal interactions from video records (see “Methods”). Teams that manage to escape within one hour are deemed ’successful’, while teams unable to do so are labeled as ‘failed’ groups. Teams must explore and exploit information, possibly through collaborations, by searching for clues, deciphering codes or opening locks. None of these tasks require any specific knowledge, skills or capabilities. However, experienced escape room players can be expected to have an advantage. Therefore, all teams in our data contain inexperienced members only (first-timers, or people who played maximum once in another room). For each team, we also recorded the sociodemographic characteristics of each player (such as age, gender and education), and relational data among team members (such as prior acquaintanceship and meeting frequency), by a questionnaire filled individually right before the game.

Our data can be mapped as a temporal network, where interactions among individuals occur at specific points in time43. In Fig. 1 we present a schematic picture of collaborative social activities in escape rooms and its network representation at five different temporal snapshots. We record both verbal and non-verbal (e.g. showing something) interaction ties between team members, between one sender and one receiver (pairwise interaction) or more than one receiver (group interaction). Interactions are directed, and assigned either a neutral, positive or negative sign (see “Methods”). Summary statistics of the investigated groups are reported in the  Supplementary Material. In the following, we provide a data-driven characterization of different dimensions of the social dynamics taking place in escape rooms.

### General features of collaborative groups

Conversations in escape rooms are fast-paced, with teams having on average 30 interactions per minute among members, each one typically lasting 3 s. We find that older and more educated players speak for longer stretches at a time (Fig. 2A,B). The strength of a relationship between individuals is associated with the intensity of their collaborative pairwise interactions during problem-solving. In particular, prior member familiarity seems to promote communication, and the more time two players have known each other, the higher their rate of communications (Fig. 2C). Moreover, people who meet frequently (more than once in a month) interact approximately 1.6 times more during problem-solving than those with lower meeting frequency (Fig. 2D).

### The microscopic architecture of team interactions

Conversations involve participants taking turns to speak, punctuated by stretches of silence. Changes in these turns among the participants are governed by rules, ensuring a basic level of order and intelligibility. To understand which rules govern collaborative communication in escape rooms, we use the Participation shift profile (P-shifts) framework developed by Gibson44. P-shifts are behaviourally meaningful and easily interpretable inventory of network motifs, small subgroup patterns that carry information about the underlying mechanisms of social interactions. We count the frequency of all possible P-shifts associated with pairwise interactions, capturing the rules of conversation a team adopts. P-shifts can be categorized into three types—turn-receiving, turn-continuing and turn-usurping—for interpretation purposes (Fig. 3A), on which we elaborate in the following. By comparing motif counts with what is found in a suitable null model (see “Methods”), we determine whether the motif frequencies of the empirical data are significantly different from those observed by random chance in systems that preserve the same number of interactions, but where temporal correlations are washed away (Fig. 3B). We apply a 5 s upper bound as the maximum allowed timegap between successive interactions to be considered a valid P-shift (results do not depend significantly on the exact choice of this parameter).

The most frequent P-shift is AB-BA, a turn-receiving shift, where the second speaker B receives the entitlement to speak from the first speaker A. For this reason, the AB-BA motif is often referred to as the “current-select” rule, and usually covers questions, commands, requests or accusation-denial type of action-reaction pairs. The over-representation of this motif suggests a high demand for practical actions and task implementation under time pressure in escape rooms. Another turn-receiving motif, AB-BY, was instead significantly underrepresented in our data. AB-BY refers to those situations when an unaddressed recipient is turned into a target in the next speaking turn. This interaction chain is rare, in contrast to AB-BA, indicating that escape room players tend to have repeated exchanges as pairs, possibly associated with the presence of subgroups within teams. Turn continuing motifs (such as AB-A0 and AB-AY) where the same speaker shifts from one target to another, either a group or a third person, are also found to be statistically underrepresented.

Interestingly, a few roles tend to be associated with particular socio-demographic characteristics, such as age, education and gender, highlighting—in Gibson’s words—a role differentiation44. This can be easily quantified by computing the Spearman’s rank correlation $$r_s$$ and the Kolmogorov–Smirnov 2-sample test statistic D between a specific P-shifts motif and such features (see “Methods” for details). We find that older people are more inclined to address the group regardless whether they receive the turn (as B in AB-B0, $$r_s=0.17$$, $$p_{val}=0.025$$), or usurp the turn (as X in AB-X0, $$r_s=0.2$$, $$p_{val}=0.025$$). Players with the lowest level of education (elementary school) are more inclined to assume turn-usurper roles (both as X in AB-XA, $$r_s=-0.2$$, $$p_{val}=0.007$$, and in AB-XB, $$r_s=-0.16$$, $$p_{val}=0.037$$), though they are less likely to turn-usurp by addressing the group (AB-X0, $$r_s=0.16$$, $$p_{val}=0.031$$).

We also observe correlations between gender and given roles in escape rooms. We find that women typically turn usurp (as X in AB-X0/AB-XA/AB-XB/AB-XY, $$D=0.23$$, $$p_{val}=0.02$$, Fig. 3C) at the expense of receiving the turn to speak (as B in AB-BA/AB-B0/AB-BY, $$D=0.23$$, $$p_{val}=0.019$$, Fig. 3D). This shows that women are rarely given the opportunity to speak after being addressed, and that they have to rather seize the floor in order to express their opinion. Indeed, these associations are even more pronounced ($$D=0.61$$, $$p_{val}=0.039$$ and $$D=0.78$$, $$p_{val}=0.003$$, respectively) when we reduced our analysis to male-dominated teams only, where there is a single woman in the team. We also observe that the lack of turn receiving opportunities lead women in male-dominated groups to communicate much less than their teammates (as measured by their fraction of outgoing communication ties, $$D=0.611$$, $$p_{val}=0.04$$), a finding which is neither present in balanced groups nor in women dominated groups. This suggests that the social environment can impose constraints on women communication and relationship behavior in task-focused groups, complementing previous observations45.

### Signed interactions

Our experiment also gave us access to information about the emotional load of each interaction, classified as neutral, positive or negative. Praising, encouragement, and in general, every relationship-oriented behavior that has a positive effect on team spirit was classified as a positive tie. Negative edges were considered those that influence team spirit adversely such as creating tension, provocation or disparagement. The wide majority of the recorded task-related exchanges were classified as neutral. However, if a task-related remark is emotionally loaded (e.g. somebody is yelling), a non-neutral sign is assigned to the communication tie (negative). This classification aims to mimic the well-established distinction between task-related and relationship-related social behavior46,47,48. More details can be found in the “Methods” section.

We find that both older (Fig. 4A) and more educated players tend to have a smaller amount of emotionally loaded interactions ($$r_s=-0.27$$, $$p_{val}=0.0003$$ and $$r_s=-0.18$$, $$p_{val}=0.021$$, respectively). By looking at their signs, we observe that the fraction of negative ties are the ones that account for the previous, negative correlations ($$r_s=-0.3$$, $$p_{val}=0.00007$$ and $$r_s=-0.23$$, $$p_{val}=0.003$$, for age and education level, respectively), while these two socio-demograhic features are not associated with higher chances of initiating positive interactions. These associations suggest that older and more highly educated team members are less likely to engage in social behavior that has a negative effect on team spirit.

Although there is a similar amount of emotional interactions in both successful and failed teams, Fig. 4B suggests that these emotional interactions are not equally distributed. Successful teams are more balanced in terms of emotional charge ($$D=0.40$$, $$p_{val}=0.078$$). In other words, members of successful teams tend to initiate a similar amount of emotional interactions with one another, while in failed groups an emotional polarization emerges, where only one or two actors display a higher number of emotionally loaded behavior. An investigation of how emotional ties evolve across time during the task performance reveals different temporal patterns for successful and failed teams (Fig. 4C). During the initial stages of the game, failed teams show a higher rate of emotionally loaded interactions. However, over time the rate of non-neutral interactions becomes higher for successful teams.

We can gain more insights into these patterns by decomposing signed interactions into positive and negative ties (Fig. 4D). Surprisingly, the initially greater amount of emotionally loaded interactions in failed teams is due to a high rate of positive interactions ($$13\%$$, doubling the number of successful teams), which, however rapidly declines within the first 15 min. This unexpected feature might reflect a lower focus (e.g. making jokes, laughing), which will eventually reveal crucial for the outcome of the game. As the game progresses, we find a greater frequency of positive interactions for successful teams, reaching $$14\%$$ after 40 min, when many crucial tasks have already been completed. The dynamics of negative interactions is less rich. Negative ties are less frequent than positive ones for both successful and failed teams, and show no significant temporal patterns, with the exception of the rapid decline in negative interactions for successful teams towards the very end of the game. The slightly greater rate of negative ties in successful teams compared to failed ones ($$6\%$$ and $$4\%$$ respectively) suggests that a minimum level of tense communication might be beneficial for task performance in collective problem solving. As some successful teams start leaving the room around the 47 min mark, only the remaining successful teams contribute to the curves shown in Fig. 4C,D.

### Group interactions

Beyond pairwise communication, we also investigate group interactions49, where more than a single recipient is addressed. Group interactions constitute $$\approx 20 \%$$ of all the interactions (see  Supplementary Material for more information), are equally divided into interactions involving 3 and 4 individuals together, and their duration is shorter, typically 77% of one-to-one interactions. We find that older (Fig. 5A) and more educated (Fig. 5B) people have a higher frequency of group interactions ($$r_s=0.23$$, $$p_{val}=0.003$$, and $$r_s=0.2$$, $$p_{val}=0.009$$, respectively), suggesting an unequal level of confidence and authority in the teams. Women display homophily when interact in groups (Fig. 5C), a phenomenon not observed when considering pairwise interactions only. No other sociodemographic data was found to be positively correlated with the presence of group interactions. All such analyses have been carried out by comparing the homophilic patterns observed in the real data with what is expected in corresponding null models where both group compositions as well as the number of interactions of each individual member is preserved.

Observing the temporal dynamics (Fig. 5D), teams tend to start the game with a high amount of group interactions. This pattern is associated with teams ‘warming up’, familiarizing themselves with the environment, and discussing together a strategy. Such a number falls abruptly after the first 5 min, reflecting division of work and individual focus. Such temporal group dynamics also display differences between successful and failed teams. Failed teams show a greater amount of group interactions around the 20-minute mark, associated with the emergence of potential early problems. By contrast, successful teams only display a low rate of group interactions around the same time, possibly reflecting greater focus and productivity. After manual visual inspection from the videos, we were able to associate the peak in group interactions in successful teams at the 30-minute mark to the presence of productive halftime get-together, where members synthesize their knowledge, discuss achievements how to proceed. As a few successful teams managed to escape slightly before the end of the game, only the remaining successful teams contribute to the curves shown in Fig. 5D.

Conversation rules governing group interactions can be analysed by a suitable extension of the previously introduced P-shift profiles44. Following a group remark, where 0 identifies the group, either the speaker continues their turn (as A in A0-AY, turn continuing), or someone else claims the turn (turn claiming). Turn-claiming can happen in three distinct ways, as X in A0-XA/A0-X0/A0-XY, where the addressee can be the first speaker A, the group 0, or another person Y. Observing the pattern of ordered group interactions accounting for conversation rules, we find that when someone claims the floor, it typically happens in the form of A0-X0 or A0-XA at the expense of A0-XY (Fig. 5E). These two P-shifts, overrepresented in our data, cover those situations when a remark addressed to the group is followed by a reaction.

To assess the relevance of each motif, we compare its frequency in the real data against what observed by random chance in systems which preserve the same number of interactions but where temporal correlations have been eliminated (Fig. 5A, no significant difference between successful and failed teams). We find that the most overabundant motif is A0-XA, the ’group version’ of the previously discussed AB-BA pairwise motif44. A second group motif, A0-X0, is also found to be overabundant. Both of these P-shifts (someone addressing a group followed either by a second speaker addressing the group again or a team representative replying to the original communicator) are usually associated with the delivery of complex messages. For example, after A proposes an idea or gives an instruction to the group, X takes the floor to explain or translate A’s idea (A0-X0), or asks for clarification to A’s action on behalf of the team (A0-XA).

## Discussion

Despite recent intensive efforts in characterizing human behavior from large-scale data, understanding the social and demographic drivers of successful team interactions is still a largely open and widely debated research area. Here we provided for the first time a characterization of the social dynamics of team interactions in escape rooms, non-interventional social laboratories previously unexplored in a fine-grained quantitative manner. By capturing high-frequency real-time social interactions in this innovative quasi-experimental setting, we were able to extract the building blocks of cooperative work. Our analysis revealed that socio-demographic characteristics might impact problem-solving communication. For instance, older and highly educated actors were observed to speak for longer, more often to the group, and initiate less emotional (in particular less negative) interactions, while prior strength of relationships between group members was positively associated with enhanced social interaction.

An investigation of P-shift profiles revealed the high demand for practical actions under time pressure, manifested in conversation rules such as the “current-select” rule (AB-BA), associated with frequent pairwise action-reaction exchanges. Interestingly, the behavior of men and women was found to be characterized by different conversation rules, with women often forced to turn-usurp in order to express their opinion, a pattern particularly overabundant in teams with only one female member. Successful teams displayed a higher emotional balance across their members, possibly reflecting the higher collective nature of team organization. In spite of a tendency for prosocial communication at the initial stage of the game, a temporal analysis revealed that already after 20 minutes failed teams had different interaction patterns from successful ones, displaying less task-focused behavior and the first signs of social conflict. Interestingly, also successful groups were found to maintain a non-negligible number of negative interactions until the end of the game, suggesting that a minimum level of tense communication might be beneficial for collective problem solving.

In summary, here we have proposed escape rooms as an innovative research setting for studying groups in a controlled, yet non-manipulated environment, where one can obtain high-resolution data on collective behavior. By investigating social dynamics at a fine-grained scale, we were able to portray an innovative and nuanced picture of the collective actions of these project teams. In the future, we intend to further investigate the division of work in these problem-solving groups to understand how exploration and exploitation tasks impose different demands on teams to allocate their resources, and how they are associated with bottlenecks slowing down collective performance. We hope that these insights will spark more research on team processes in escape rooms, easily accessible social laboratories contributing to a deeper understanding of intra-group dynamics and collective action.

## Methods

### Data processing

Interaction data on the problem-solving activity of teams were extracted from the video records by selected trained transcribers who were trained through several individual sessions. As a test of inter-coder reliability, following individual training, all applicants received the same video to code as a trial. In particular, consistency among and within transcribers was quantified through the Krippendorff alpha test, for which we obtained a value of 0.73, well above the conventional threshold of 0.6550. After the transcribers selection was completed, randomly chosen samples of the transcriptions were double-checked in a systematic manner, obtaining similar consistency values. Transcribers were given a detailed guide on how to translate video information into a temporal signed edge list. Above all, transcribers had to consider that an interaction covers both verbal (one line of thought with maximum a few seconds of silence) and non-verbal communication (e.g., nodding). The start time and end time of an interaction is the second where the given communication is already/still present.

### Signed interactions

We categorized the emotional load of the interactions and associated to each tie either a positive, negative or neutral sign. A precise guide on the classification of signed interactions was shared with the transcribers. In particular, positive interactions are all interactions that relate to the previous action positively/have positive emotional charge and is not directed to the task itself but is rather related to interpersonal relationships. Typical examples of positive interactions are praising, laughing, joking and the encouragement/provision of emotional support of one another. In sum, all communication that are expected to influence the unity and spirit of the team positively is recorded as a positive interaction. In contrast, disparagement, despising, ironic, cynical reaction, provocation, intentional ignoring of someone, and in general, any other interaction creating tension between group members are considered negative. Therefore, negative interactions are those communication ties that relate to the previous action negatively and are not directed to the task, and thereby, they are assumed to influence team spirit and unity negatively. Task-related interactions, no matter if they express agreement or disagreement, are classified as neutral, as soon as they do not share any of the features described above, i.e. they do not carry emotional load. Typical neutral communication ties are task-related questions, replies and simple disclosure of facts. Taken together, neutral interactions cover all emotionally indifferent, task-focused communication, while signed interactions are those influencing team spirit positively/negatively, and which are constructive/disruptive to social relations in the context of a collaboration-demanding, pressed environment. In addition to this information, transcribers were given specific examples taken from the videos on managing correctly the categorization of signed interactions. Inter-coder consistency for signed interactions was evaluated as part of the consistency test described above. A 5-min rolling window was applied for plots showing the temporal evolution of signed interactions (Fig. 4C,D).

### Group interactions

When three or more individuals were found to interact together, all actors involved were considered members of the same group, encoding communication beyond dyadic ties and traditional links. A detailed review of the formalism and mathematical tools used to properly encode group interactions can be found in Ref.49. The ratio between the number of group interactions and the pairwise one can be computed by dividing the total number of hyperedges of order greater than two by the number of hyperedges of order equal to two. More information on the sociodemographic compositions of groups can be found in the  Supplementary Material. A 5-min rolling window was applied for plots showing the temporal evolution of groups (Fig. 5D).

### Null model for identifying P-shifts

To capture P-shift motifs observed in team communication patterns, we compare the abundance of each motif in the real data with what is expected in a suitable null model. In particular, we take the time series of interactions within a team during the play and obtain a randomized version of it by shuffling the order in which these interactions occur. This kind of randomization preserves the total interactions initiated/received by each individual, as well as the frequency and composition of group interactions. However, the temporal order of events is completely randomized. Any deviation from this null model signals the presence of temporal correlations in form of individual memory of past contacts, which do not merely depend on the activity and the total number of interactions of each individual.

### Statistical tests

Spearman’s correlation coefficient is a nonparametric measure of rank correlation. For this reason, it is able to assess the presence of positive monotonic relationships, whether they are linear or not. If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other. The significance level of the found correlation is described by the corresponding p-value.

Kolmogorov–Smirnov (KS) 2-sample test is a nonparametric test typically used to assess whether two underlying one-dimensional probability distributions differ in a significant way. In particular, the KS test statistic is defined as $$D_{m,n}$$ = $$max_x |F(x)-G(x)|$$, where the two samples have sizes mn, with observed cumulative distribution functions (CDFs) of F(x), G(x) respectively. The null hypothesis i.e. both samples come from the same distribution, can be rejected (at significance level $$\alpha$$) if $$D_{m,n} > D_{m,n,\alpha }$$ where the critical value $$D_{m,n,\alpha }$$ can be found online in the corresponding standard table.

### Ethical statements

The authors confirm that all experimental protocols were approved by the Ethical Research Committee of Central European University. We confirm that all methods were carried out in accordance with the Committee’s Ethical Research Policy and Guidelines (https://acro.ceu.edu/sites/acro.ceu.edu/files/attachment/basicpage/141/ethicalresearchpolicyandguidelines_0.pdf) including obtainment of informed consent from all participants.

## Data availability

Data are available at: https://github.com/bambuszka/escape_room.

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## Acknowledgements

R.O.S. and F.B. wish to thank to Balázs Lengyel and László Lőrincz for their feedback, and Szabolcs Tóth-Zs for his help producing the visualizations. R.O.S. also acknowledges useful conversations with Adam R. Pah. This publication is the result of research conducted for Central European University, Private University – CEU GmbH. It was made possible by the CEU Open Access Fund.

## Author information

Authors

### Contributions

R.O.S. and F.B. designed research. R.O.S. designed the surveys, organized the experimental set-up and conducted the data collection. R.O.S. and S.C. analyzed the data, with support from D.D. R.O.S. and F.B. wrote the paper, with contributions from S.C. and D.D.

### Corresponding authors

Correspondence to Rebeka O. Szabo or Federico Battiston.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

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O. Szabo, R., Chowdhary, S., Deritei, D. et al. The anatomy of social dynamics in escape rooms. Sci Rep 12, 10498 (2022). https://doi.org/10.1038/s41598-022-13929-0

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• DOI: https://doi.org/10.1038/s41598-022-13929-0


text:
Violation of the Newton's first law of motion in the movement of spacecrafts in the vacuum of space [closed]

Suppose a rocket at rest in space where there is a complete vacuum and no force to influence the rocket:

In the left we have the rocket with a metal ball in it. After the ball is thrown out of the rocket, according to Newton's first law of motion, because no force is influencing the system ($\sum F = 0)$ and the system is initially at rest, the center of mass of the system will continue to be at rest, although the ball and the rocket will move in opposite directions. In this case we may reduce the Newton's first law of motion to this equation:

$$m_1 v_1 = m_2 v_2$$
(as it might be interpreted as the conservation of momentum)

Thus, dependent on the mass of the rocket $(m_1)$ and the mass of the metal ball $(m_2)$ and the velocity of the ball when moving out of the rocket $(v_2)$, we can calculate the velocity of the rocket $(v_1)$.

Now, in the right we have the same rocket but this time with a tank of liquefied gas (a real gas) instead of the metal ball. After opening the tank the gas will flow out of the rocket in the vacuum and we expect the rocket to move in the opposite direction the same way jet spacecrafts allegedly move in the vacuum of space. Suppose the mass of the liquefied gas initially is again $m_2$ and suppose after the gas flows out of the tank completely the rocket gets the velocity of $v_1$ and the mass of the rocket(including the empty tank) is again $m_1$. Applying the Newton's first law of motion again in this case we are (or should be) able to calculate the mean velocity $(v_2)$ of the total mass of the gas that have been flowed and scattered in the vacuum, HOWEVER obviously it is impossible to logically think of such a velocity! The molecules of the gas are moving in any direction in the vacuum of space; they won't move in the space the same way that a metal ball will move. In other words we can't THROW a mass of gas in the space and get some movement in the opposite direction out of it.

Conclusion: Movement of a jet spacecraft in the vacuum of space is impossible!

Question: What's wrong with the above reasoning?

closed as unclear what you're asking by CuriousOne, Daniel Griscom, Ali, yuggib, Kyle KanosJan 14 '16 at 11:05

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• You went wrong with: "The molecules of the gas are moving in any direction in the vacuum of space; they won't move in the space the same way that a metal ball will move." I'm not sure why you assume that the gas from a rocket sprays out in all directions. Maybe you are picturing the turbulence exhibited by rocket exhaust when the rocket is first leaving the launch pad? – mbeckish Jan 13 '16 at 18:07
• "HOWEVER" (not even capitalized) and "obviously" are not logical statements. Using them together isn't one, either. In space the center of gravity of a rocket always stays the same. The rocket body moves one way, the propellant another and the center of gravity stays where it was (or moves at a constant velocity in your choice of inertial system). Can you calculate the center of gravity of a rocket body and a gas cloud? Yes. – CuriousOne Jan 13 '16 at 18:07
• Thanks for your comment > Curious That gas cloud that you say is just a mental imagination. You are thinking of that gas cloud just the same as you think of a cloud in the earth's atmosphere. We are talking about vacuum in which you may never have such a cloud of gas. The expansion of that gas cloud to the space infinity will be immediate. after a few moments you may never have such cloud as you may have in an atmosphere. – mithridates2000 Jan 13 '16 at 18:18
• mbeckish: Here we are talking about the moment when the gas is completely flowed in the vacuum and the moments after that. Wouldn't you have the total mass of gas scattered all in the space? Can you think of a gas cloud that is moving with the velocity of v2 as Curious mentioned it? – mithridates2000 Jan 13 '16 at 18:25
• Gas doesn't expand to infinity immediately. – CuriousOne Jan 13 '16 at 19:37