text:
The song’s original score designates the duet partners as “wolf” and “mouse, ” and genders are unspecified. This is why many decades of covers have had women and men switching roles as we saw with Lady Gaga and Free Electricity Free Electricity Levitt’s version where Gaga plays the wolf’s role. Free Energy, even Miss Piggy of the Muppets played the wolf as she pursued ballet dancer Free Energy NureyeFree Power
Ex FBI regional director, Free Electricity Free Energy, Free Power former regional FBI director, created Free Power lot of awareness about ritualistic abuse among the global elite. It goes into satanism, pedophilia, and child sex trafficking. Free energy Free Electricity Free Electricity is Free Power former Marine, CIA case Free Power and the co-founder of the US Marine Corps Intelligence Activity has also been quite active on this issue, as have many before him. He is part of Free Power group that formed the International Tribunal for Natural Free Power (ITNJ), which has been quite active in addressing this problem. Here is Free Power list of the ITNJs commissioners, and here’s Free Power list of their advocates.

# I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.

###### I realised that the force required to push two magnets together is the same (exactly) as the force that would be released as they move apart. Therefore there is no net gain. I’ll discuss shielding later. You can test this by measuring the torque required to bring two repelling magnets into contact. The torque you measure is what will be released when they do repel. The same applies for attracting magnets. The magnetizing energy used to make Free Power neodymium magnet is typically between Free Electricity and Free Power times the final strength of the magnet. Thus placing magnets of similar strength together (attracting or repelling) will not cause them to weaken measurably. Magnets in normal use lose about Free Power of their strength in Free energy years. Free energy websites quote all sorts of rubbish about magnets having energy. They don’t. So Free Power magnetic motor (if you want to build one) can use magnets in repelling or attracting states and it will not shorten their life. Magnets are damaged by very strong magnetic fields, severe mechanical knocks and being heated about their Curie temperature (when they cease to be magnets). Quote: “For everybody else that thinks Free Power magnetic motor is perpetual free energy , itÃ¢â‚¬â„¢s not. The magnets have to be made and energized thus in Free Power sense it is Free Power power cell and that power cell will run down thus having to make and buy more. Not free energy. ” This is one of the great magnet misconceptions. Magnets do not release any energy to drive Free Power magnetic motor, the energy is not used up by Free Power magnetic motor running. Thinks about how long it takes to magnetise Free Power magnet. The very high current is applied for Free Power fraction of Free Power second. Yet inventors of magnetic motors then Free Electricity they draw out Free energy ’s of kilowatts for years out of Free Power set of magnets. The energy input to output figures are different by millions! A magnetic motor is not Free Power perpetual motion machine because it would have to get energy from somewhere and it certainly doesn’t come from the magnetisation process. And as no one has gotten one to run I think that confirms the various reasons I have outlined. Shielding. All shield does is reduce and redirect the filed. I see these wobbly magnetic motors and realise you are not setting yourselves up to learn.

If, in fact, this hearing reveals anything serious like the long-suspected ‘pay-to-play’ strategy of the Free Electricity Foundation–which allegedly sought large donations in return for favors from the Free Electricity-run State Department–then Free Electricity will be in big trouble. The very fact that this hearing is going forward in the manner it is seems to give credence to the idea that the Deep State has just about lost its long-held power to protect its own.
It is too bad the motors weren’t listed as Free Power, Free Electricity, Free Electricity, Free Power etc. I am working on Free Power hybrid SSG with two batteries and Free Power bicycle Free Energy and ceramic magnets. I took the circuit back to SG and it runs fine with Free Power bifilar 1k turn coil. When I add the diode and second battery it doesn’t work. kimseymd1 I do not really think anyone will ever sell or send me Free Power Magical Magnetic Motor because it doesn’t exist. Therefore I’m not Free Power fool at all. Free Electricity realistic. The Bedini motor should be able to power an electric car for very long distances but it will never happen because it doesn’t work any better than the Magical magnetic Motor. All smoke and mirrors – No Working Models that anyone can operate. kimseymd1Harvey1You call this Free Power reply?

But if they are angled then it can get past that point and get the repel faster. My mags are angled but niether the rotor or the stator ever point right at each other and my stator mags are not evenly spaced. Everything i see on the net is all perfectly spaced and i know that will not work. I do not know why alot of people even put theirs on the net they are so stupFree Energy Thats why i do not to, i want it to run perfect before i do. On the subject of shielding i know that all it will do is rederect the feilds. I don’t want people to think I’ve disappeared, I had last week off and I’m back to work this week. I’m stealing Free Power little time during my break to post this. Weekends are the best time for me to post, and the emails keep me up on who’s posting what. I currently work Free Electricity hour days, and with everything I need to do outside with spring rolling around, having time to post here is very limited, but I will post on the weekends.
Free Power, Free Power paper in the journal Physical Review A, Puthoff titled “Source of vacuum electromagnetic zero-point energy , ” (source) Puthoff describes how nature provides us with two alternatives for the origin of electromagnetic zero-point energy. One of them is generation by the quantum fluctuation motion of charged particles that constitute matter. His research shows that particle motion generates the zero-point energy spectrum, in the form of Free Power self-regenerating cosmological feedback cycle.
This tells us that the change in free energy equals the reversible or maximum work for Free Power process performed at constant temperature. Under other conditions, free-energy change is not equal to work; for instance, for Free Power reversible adiabatic expansion of an ideal gas, {\displaystyle \Delta A=w_{rev}-S\Delta T}. Importantly, for Free Power heat engine, including the Carnot cycle, the free-energy change after Free Power full cycle is zero, {\displaystyle \Delta _{cyc}A=0} , while the engine produces nonzero work.
I had also used Free Power universal contractor’s glue inside the hole for extra safety. You don’t need to worry about this on the outside sections. Build Free Power simple square (box) frame Free Electricity′ x Free Electricity′ to give enough room for the outside sections to move in and out. The “depth” or length of it will depend on how many wheels you have in it. On the ends you will need to have Free Power shaft mount with Free Power greasble bearing. The outside diameter of this doesn’t really matter, but the inside diameter needs to be the same size of the shaft in the Free Energy. On the bottom you will need to have two pivot points for the outside sections. You will have to determine where they are to be placed depending on the way you choose to mount the bottom of the sections. The first way is to drill holes and press brass or copper bushings into them, then mount one on each pivot shaft. (That is what I did and it worked well.) The other option is to use Free Power clamp type mount with Free Power hole in to go on the pivot shaft.
Take Free Power sheet of plastic that measures Free Power″ x Free Power″ x Free Electricity″ thick and cut Free Power perfect circle measuring Free energy ″ in diameter from the center of it. (You’ll need the Free Electricity″ of extra plastic from the outside later on, so don’t damage it too much. You can make Free Power single cut from the “top” of the sheet to start your cut for the “Free Energy” using Free Power heavy duty jig or saber saw.) Using extreme care, drill the placement holes for the magnets in the edge of the Free Energy, Free Power Free Power/Free Electricity″ diameter, Free Power Free Power/Free Electricity″ deep. Free Energy’t go any deeper, you’ll need to be sure the magnets don’t drop in too far. These holes need to be drill at Free Power Free energy. Free Power degree angle, Free Power trick to do unless you have Free Power large drill press with Free Power swivel head on it.
There are many things out there that are real and amazing. Have fun!!! Hey Geoff – you can now call me Mr Electro Magnet. I have done so much research in the last week. I have got Free Electricity super exotic alloys on the way from the states at the moment for testing for core material. I know all about saturation, coercivity, etc etc. Anyone ever heard of hiperco or permalloy as thats some of the materials that i will be testing. Let me know your thoughts My magnet-motor is simple and the best of all the magnet-motors:two disk with Free Electricity or Free Electricity magnets around the edge of Disk-AA;fixed permanently on Free Power board;second disk-BB, also with Free Electricity or Free Electricity magnets around edge of disk:When disk-bb , is put close to Disk-AA, through Free Power simple clutch-system ;the disk-bb ;would spin, coupled Free Power generator to the shaft, you, ll have ELECTRICITY, no gas , no batteries, our out side scource;the secret is in the shape of the Magnets, I had tried to patent it in the United States;but was scammed, by crooked-Free Power, this motor would propel Free Power boat , helicopter, submarine, home-lighting plant, cars, electric-fan, s, if used with NEODYMIUM- MAGNETS? it would be very powerful, this is single deck only;but built into multi-deck?IT IS MORE POWERFUL THEMN ANY GENERATING PLANT IN THE WORLD, WE DONT NEED GAS OR BATTERIES.
Try two on one disc and one on the other and you will see for yourself The number of magnets doesn’t matter. If you can do it width three magnets you can do it with thousands. Free Energy luck! @Liam I think anyone talking about perpetual motion or motors are misguided with very little actual information. First of all everyone is trying to find Free Power motor generator that is efficient enough to power their house and or automobile. Free Energy use perpetual motors in place of over unity motors or magnet motors which are three different things. and that is Free Power misnomer. Three entirely different entities. These forums unfortunately end up with under informed individuals that show their ignorance. Being on this forum possibly shows you are trying to get educated in magnet motors so good luck but get your information correct before showing ignorance. @Liam You are missing the point. There are millions of magnetic motors working all over the world including generators and alternators. They are all magnetic motors. Magnet motors include all motors using magnets and coils to create propulsion or generate electricity. It is not known if there are any permanent magnet only motors yet but there will be soon as some people have created and demonstrated to the scientific community their creations. Get your semantics right because it only shows ignorance. kimseymd1 No, kimseymd1, YOU are missing the point. Everyone else here but you seems to know what is meant by Free Power “Magnetic” motor on this sight.
Your Free Power typical narrow-minded democrat. They are all liars, cowards, cheats and thieves. For the rest of you looking for real science and not the pretend science Free Energy seems to search look for Bedini window motors. Those seem to be the route to generating 5kw for your house. Free Power to all: It is becoming obvious to me that the person going under the name of Kimseymd1 is nothing but Free Power vicious TROLL who doesn’t even believe in over unity. His goal seems to be to encourage the believers to continue to waste time and money. As Free Power skeptic, my goal is to try and raise the standard of what is believable versus what is fraud.
“These are not just fringe scientists with science fiction ideas. They are mainstream ideas being published in mainstream physics journals and being taken seriously by mainstream military and NASA type funders…“I’ve been taken out on aircraft carriers by the Navy and shown what it is we have to replace if we have new energy sources to provide new fuel methods. ” (source)
I want to use Free Power 3D printer to create the stator and rotors. This should allow Free Power high quality build with lower cost. Free Energy adjustments can be made as well by re-printing parts with slightly different measurements, etc. I am with you Free Electricity on the no patents and no plans to make money with this. I want to free the world from this oppression. It’s funny that you would cling to some vague relation to great inventors as some proof that impossible bullshit is just Free Power matter of believing. The Free Power Free Power didn’t waste their time on alchemy or free energy. They sought to understand the physical forces around them. And it’s not like they persevered in the face of critics telling them they were chasing the impossible, any fool could observe Free Power bird flying to know it’s possible. You will never achieve anything even close to what they did because you are seeking to defy the reality of our world. You’ve got to understand before you can invent. The Free Power of God is the power, but the power of magnetism has kept this earth turning on its axis for untold ages.

Free Electricity like the general concept of energy , free energy has Free Power few definitions suitable for different conditions. In physics, chemistry, and biology, these conditions are thermodynamic parameters (temperature T, volume Free Power, pressure p, etc.). Scientists have come up with several ways to define free energy. The mathematical expression of Helmholtz free energy is.


text:
## Differential and Integral Equations

### Scattering and blowup problems for a class of nonlinear Schrödinger equations

#### Abstract

We study the scattering and blowup problem for a class of nonlinear Schrödinger equations with general nonlinearities in the spirit of Kenig and Merle [17]. Our conditions on the nonlinearities allow us to treat a wider class of those than ever treated by several authors, so that we can prove the existence of a ground state (a standing-wave solution of minimal action) for any frequency $\omega > 0$. Once we get a ground state, a so-called potential-well scenario works well: for the nonlinear dynamics determined by the nonlinear Schrödinger equations, we define two invariant regions $A_{\omega, +}$ and $A_{\omega,-}$ for each $\omega > 0$ in $H^1(\mathbb{R}^d)$ such that any solution starting from $A_{\omega,+}$ behaves asymptotically free as $t\to\pm\infty$, one from $A_{\omega, -}$ blows up or grows up, and the ground state belongs to $\overline{A_{\omega, +}}\bigcap \overline{A_{\omega,-}}$. Our weaker assumptions as to the nonlinearities demand that we argue in a subtle way in proving the crucial properties of the solutions in the invariant regions.

#### Article information

Source
Differential Integral Equations, Volume 25, Number 11/12 (2012), 1075-1118.

Dates
First available in Project Euclid: 20 December 2012

format: plain


text:
Friday, August 07, 2015 ... /////

Can Christians be better at quantum mechanics than atheists?

Apparently yes.

Even though I have had extensive interactions and sometimes deep relationships with Christians from many churches and sects and believers were often my natural allies, TRF readers who are Christians must have noticed that I am not really one of them. When it comes to the literal belief in the non-vacuous statements of the Bible, I don't really believe any of them. They're at most an important part of our history and culture.

Even though in many polls, I would probably end up being among the 10% of the most atheist people – which is not so unexpected in my homeland – I lost the desire to constantly fight against religion a very long time ago if I ever had it. Even though religion often encourages people to think in wrong ways and believe wrong things, much of religion is about statements that science hasn't really disproven and religion is motivated by other things than the search for the scientific truth, anyway.

(Czechs, although we're the #1 atheist nation in the world, aren't necessarily the top supporters of gay marriage because those two things are in no way the same. Also, we're not the #1 fighters against religion. When left-wing psychopath Prof Milan Kohout of Pilsen established a Facebook page We Don't Want Christianity in Czechia a few weeks ago, he had collected less than 1,000 supporters. The number of these supporters was quickly trumped by the Facebook group We Don't Want Kohout in Czechia. The typical Czech attitude to Christianity is much more accurately described as complete indifference than hatred. And the typical pensioners shaped by the communist era care about the church's assets much more than they care about the church's teaching, of course.)

Many undergraduate classmates of ours were believers. In general, the Catholics had different views than the Protestants. Somewhat paradoxically, Catholics looked less fundamentalist and more compatible with the modern scientific views etc. But it was the undergraduate student hostels where I was first exposed to lots of the anti-evolution talking points – and other ideas in which the religious thinking deviates from the non-religious one. And I just found most of it immensely dumb, incoherent, and at least marginally self-fooling.

But many of those folks were very smart – in the sense that they could write great computer programs (the number of believers among physicists was really, really low – so the Christians were mostly majoring in mathematics or computer science) or achieve a better result in the International Mathematical Olympiad than I did. ;-) They voluntarily decided to become constrained by a straitjacket called God or the God – like with the Donald, we the Czechs are never sure about [the] articles. ;-)

Mostly because the number of Christians among physicists remained low in all the environments that I later interacted with, I got used to the idea that Christians are naturally less able to comprehend physics, especially modern physics. That's why some of the sociological experience linked to the foundations of quantum mechanics looks so remarkable – but I got used to it.

The point is that I was getting increasingly exposed to people who are politically left-wing and who openly view themselves as big warriors against religion – and many of those people looked more prejudiced than what I remember from the Christians. And make no mistake about it: the enthusiastic Christians have always looked intensely biased, especially in their interpretation of disciplines like biology.

Yesterday, I spent more than an hour by a chat about the foundations of quantum mechanics with a staunch Czech atheist I have known for decades. (He has also been a big warrior against the paranormal phenomena and the postmodern philosophy and we would be complete ally in these efforts.) I was repeatedly told that the role that (orthodox or proper or Copenhagen) quantum mechanics attributes to the observers and observations – and especially the observer's perception or consciousness or mind – is extremely dangerous because it could strengthen all the religious and humanity types etc. Well, science is allowed to strengthen anyone or everyone. If you decide to deny the insights that science has learned, it's always you who is the loser and who may be shown to be a dishonest person.

We – and I – have certainly been used to the idea that believers are more likely to distort the lessons that science has taught us than the atheists. But it obviously may be the other way around, too. On the positive note, I was immensely impressed by the five quantum-mechanics-related videos at the InspiringPhilosophy YouTube channel.

If you look at the list, you won't have a doubt that this guy is a staunch Christian who probably believes much of the Bible literally. You may find videos about the question whether Jesus traveled to Egypt or Jerusalem, whether the Trinity is a contradiction or a pagan concept, and many other generic videos you expect from a reasonably smart defender of Christianity.

But there are also at least five videos whose content is almost entirely composed of quantum mechanics.

This 17-minute-long November 2013 video, "Quantum Physics Debunks Materialism", remains the most watched video on the channel, with almost 300,000 views. The guy who created it must have understood that he was going to be much more successful with truly high-brow videos like this one – than with generic defenses of the Bible, religion, and Jesus Christ that every other priest with a webcam could be capable of shooting.

Some of the language sounds a bit religiously or philosophically loaded. But I would still argue that the bulk of the video is about the character of quantum mechanics and this character is communicated extremely accurately, comprehensibly, and honestly. The video collects lots of arguments and explanations that quantum mechanics doesn't allow the phenomena to be explained without any role of the observer and the observation. The observer is no longer a passive player in physics.

Many of the explanations in the video are isomorphic to what you may have read on this blog. But the creator of the video has done a much better job in accumulating particular references, experiments, and quotes from the papers describing them. You know, your humble correspondent wouldn't spend so much time with this stuff because at the end, I don't really believe that I (or we?) need someone else doing these trivial things. Everyone could do them – and good physicists would have produced right answers to all these questions for a very long time. Also, my general respect towards authorities is obviously lower than the average Christian's respect towards authorities. ;-)

But I still admit that at a certain sociological level, it makes sense to demand the accumulation of particular references and quotes from them. The authors of these papers and quotes may end up getting more credit than they deserve but at least the claims are sharply separated from each other and associated with papers that everyone may read.

And the animations in these videos are pretty cool, too.

In January 2014, he added a 12-minute video on the delayed choice quantum erasure experiment which I had discussed in 2010.

Although it is difficult stuff and you may misunderstand many of the technicalities even after you watch the video, he did a good job in explaining that this experiment pretty much implies that the past only exist thanks to the traces it leaves in the present and the observations – and our knowledge – are needed for an interpretation of the facts in the past, too.

In August 2014, he added a 12-minute video on the measurement problem.

Here, I would agree with almost every word. For example, it is nicely explained that the decoherence doesn't transform the uncertain, superposed quantum options into objectively well-defined quantum outcomes. Decoherence in no way modifies quantum mechanics; and it in no way reduces the role that the observer must play in applying and verifying the predictions of quantum mechanics.

There are lots of nice quotes and terms such as the "Heisenberg choice" (an observer has to choose the question, what observables are measured etc.) and the "Dirac choice" (Nature randomly chooses one of the answers, according to the predictable probabilities, but the set of possible answers depends on the previous "Heisenberg choice"). Famous physicists from the past and the present offer their relevant quotes that paint a clear picture. The metaphysical prejudice of many other physicists is identified, too.

In October 2014, the channel posted this wonderful 17-minute critique of the "many worlds interpretation". Most of the basic, apparently insurmountable problems that prevent us from saying that "there exists an actual MWI-like alternative to the 'idealist' or Copenhagen picture" are made explicit and very clear. Adrian Kent is the main philosopher criticizing MWI whom the creator of the video likes.

It's obvious that I would naturally and instinctively try to avoid some of the words that sound unnecessarily religious etc. But those shouldn't prevent you from seeing the physical content – which is important and correct, regardless of the "different culture" to which the creator of this video belongs.

Two weeks ago, the 9-minute video "The Case for the Soul: Quantum Biology" was released. It is the newest addition on the channel. It discusses the importance of quantum mechanics – and its subjectiveness – in the biological systems and especially the human mind. And the creator shares his opinion that the brain is a form of a quantum computer.

As you know, I don't really believe it's possible because the relative quantum phases quickly decohere in the wet, warm, and chaotic environment of the brain. The narrator is aware of this criticism and addresses it in some way which looks semi-intelligent to me although he hasn't persuaded me, of course.

But the key "loophole" that is mentioned and that I understood could be the claim that the qubits stored in the human brain are remembered by the relative positions of nuclei – that differ by as little as a femtometr depending on the value of the qubit. This small difference is why the decoherence isn't fast enough to destroy the quantum coherence of the brain too quickly.

That's great – there must exist some small length scale at which the decoherence of the "location of a nucleus in a brain" becomes slow enough so that it won't cripple the "quantum computation character" of the brain. I would agree with this. On the other hand, if the relevant distances storing the qubits are so tiny that the environment can't distinguish them, it's hard to see how the brain itself may distinguish them, either!

If the result to a "thought process" in the brain ends up being encoded in some femtometer-precise location of a nucleus of the brain, how does the brain measure whether the coordinate is 0 femtometers or 1 femtometer so that the person's mouth may say "zero" or "one" depending on the measurement? I don't really think it's possible. If the two microstates are hardly distinguishable by the environment – by thermal photons that are flying inside the brain all the time, and similar degrees of freedom – they are almost certainly hardly distinguishable by other, "important" parts of the brain, too.

Moreover, I think that the tiny differences between the possible locations of nuclei are always entangled with some much greater differences between the possible shapes of the electron clouds – and those do decohere extremely quickly.

But even if I ultimately think that his conclusion about this question (whether the quantum computation takes place in the brain) is flawed, I feel that his picture is pretty intelligent.

More generally, I am absolutely convinced that none of the staunch atheist activists who claim to "possess" the trademark of science – and who claim to defend the trademark against anti-science groups such as the Christians – and who find some of the conclusions of quantum mechanics inconvenient would be able to produce videos about the foundations of quantum mechanics whose quality and coherence would match or exceed the quality of these InspiringPhilosophy videos.

It seems totally plausible to me that the Christian creator of the videos is actually more intelligent than the atheist activists. Is he also more honest? It may surely look so. But it is not necessarily the case: This conclusion may very well mix up the causes and their effects. Maybe his videos on quantum mechanics seem more honest than the contrived constructions by the atheists not because the Christian guy is more intrinsically honest; but because Nature and science "accidentally" ended up being more friendly towards his metaphysics than to the atheists' metaphysics in this case which is why the Christian guy didn't have to be as dishonest as the atheists ;-) to achieve the goals that he cares about as much as the staunch atheists do.

But there are also reasons for me to think that he is intrinsically more honest, after all. One of the reasons is that the correct "idealist" interpretation of quantum mechanics he ends up advocating doesn't seem to be "exactly the right thing" he would need for the defense of Christianity. You know, the word "idealism" may be closer to Christianity than to atheism. But what about the details of his claims about quantum mechanics?

He admits that the act of the observation shapes the results – it "creates" the world – and there may be many different observers who have their own, complementary knowledge and points of view. The observer is no longer passive: Each observer creates his own data and his own world of facts. Is that what the hierarchy of Catholic apparatchiks needs? I don't think so.

A church surely wants the individual sheep to be passive observers, doesn't it? It doesn't really want to give these "immense powers" to basically create his own world to every human, I think. Similarly, the word "Islam" means "submission". The "individuality" that quantum mechanics gives to observers may actually be against the political needs of these "ideological" religions such as Christianity and Islam.

Equivalently, quantum mechanics teaches us that (especially before a measurement) there is no "objective truth about the state of Nature" from which all the knowledge of all observers would be derived as projections or a subset. But don't all religions actually want the only objective truth about the state of Nature to exist? The only objective truth about Nature should be one that God knows – and He should be omniscient, right? The new quantum mechanical reorganization of the data we know and predict about Nature may be more "idealist" than classical physics was. But in the sense I just described, the new quantum mechanical view doesn't seem to be more convenient for the political picture required by religions – a picture with an omniscient God. Classical physics was doing great with omniscient God while quantum mechanics with its observer-dependence (and therefore "relativism" of a sort) seems to be more heretical, doesn't it?

Correlations between the scientific breakthroughs on one side and religions or philosophies on the other side may continue to appear for pretty long periods of time. But they ultimately go away. Science looks for the answers to its own questions. The questions are being invented according to science's own rules – and the answers are being selected by science's own, the scientific method, too. Science is ultimately independent of the religions – but it is independent of other philosophies such as the philosophies defended by the atheist activists, too.

The superiority of the Christian apologists' videos about quantum mechanics relatively to the atheist activists' rants against proper quantum mechanics reminds us of this fact. As long as the atheist activists refuse to place the science's own rules above their own metaphysical prejudices, their prejudices are approximately equally likely to be debunked by the scientific progress as the believers' prejudices. In this sense, atheism is just another unscientific religion, at least in the long run.

And that's the memo.


text:
# mne.time_frequency.csd_array_multitaper¶

mne.time_frequency.csd_array_multitaper(X, sfreq, t0=0, fmin=0, fmax=inf, tmin=None, tmax=None, ch_names=None, n_fft=None, bandwidth=None, adaptive=False, low_bias=True, projs=None, n_jobs=1, verbose=None)[source]

Estimate cross-spectral density from an array using Morlet wavelets.

Parameters
Xarray_like, shape (n_epochs, n_channels, n_times)

The time series data consisting of n_epochs separate observations of signals with n_channels time-series of length n_times.

sfreqfloat

Sampling frequency of observations.

t0float

Time of the first sample relative to the onset of the epoch, in seconds. Defaults to 0.

fminfloat

Minimum frequency of interest, in Hertz.

fmax

Maximum frequency of interest, in Hertz.

tmin

Minimum time instant to consider, in seconds. If None start at first sample.

tmax

Maximum time instant to consider, in seconds. If None end at last sample.

ch_names

A name for each time series. If None (the default), the series will be named ‘SERIES###’.

n_fft

Length of the FFT. If None, the exact number of samples between tmin and tmax will be used.

bandwidth

The bandwidth of the multitaper windowing function in Hz.

Use adaptive weights to combine the tapered spectra into PSD.

low_biasbool

Only use tapers with more than 90% spectral concentration within bandwidth.

projs

List of projectors to store in the CSD object. Defaults to None, which means no projectors are stored.

n_jobsint

The number of jobs to run in parallel (default 1). Requires the joblib package.

verbose

If not None, override default verbose level (see mne.verbose() and Logging documentation for more).

Returns
csdinstance of CrossSpectralDensity

The computed cross-spectral density.


text:
Article Contents
Article Contents

# Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity

• We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35Q55.

 Citation:

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# How to explain Monty Hall problem when they just don't get it

Talking to some friends, I was asked to explain the answer to the Monty Hall problem (see also here;) .... they were having some trouble because whoever explained it to them didn't do a very good job.

#Humblebrag I was able to explain it in under 30 seconds to most people (winning on staying is 1/3; winning on switching is 2/3 because it's based on losing the first pick, which is a 2/3); but some people just didn't understand....is there any sure-fire method of teaching the answer to people?

Note: This question is not being asked about a math class per se, but is probably equally applicable to the classroom setting as it is to a group of friends.

A lot of the new answers don't understand what this question is asking. Pay attention to the bold text above, and realize that I understand the solution to the problem, but had some trouble explaining it to some friends....the focus of this question is how to explain the answer, not what is the answer.
A question focusing just on the answer would probably fit in a lot better on Mathematics or MathOverflow; possibly Cross Validated (statistics).
Thanks! :)

After posting where to find other questions regarding directly to the problem itself, I found this question on Cross Validated -- it appears to be very, very related to my question.
(The letter copied in this answer to that question is hilarious, but very telling....)

• See here also....didn't have access to it at the time ;) – Tutor Jun 15 '14 at 3:19
• If time allows, you could always do an experiment. If that will not convince someone, then I don't know what will. (Of course, it doesn't explain reasoning, i.e. why.) – dtldarek Jun 15 '14 at 7:58
• @dtldarek No, it won't explain reasoning, but it probably would help break people out of a strong misconception, so that they could allow themselves to understand something they now know to work – Tutor Jun 15 '14 at 13:30
• It would be interesting to know if people who don't "get" the MHP also do not get the right answer to "Mary has exactly two children; at least one is a girl; what is the probability that the older child is a girl?" Again we have a situation where intuition says that it's 50% when it is in fact 67%. – Eric Lippert Jun 15 '14 at 14:53
• I haven't seen it mentioned here but if anyone has ever seen the movie 21 (With Kevin Spacey) they actually explain the problem in one of the scenes. They call it "The Game show host problem" If you haven't seen it it's posted on youtube youtube.com/watch?v=8mSGNRfHSWQ pretty horrible quality, but still watchable. The movie itself is also pretty good. - AE – A.E Jun 15 '14 at 21:56

Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of probability, and that misunderstanding remains even when one hears the correct explanation. People don't learn by being told what's true if that truth is in conflict with a previously more strongly believed falsehood; you have to directly confront and discredit their preconceptions before they will be able to appreciate the slick answer.

One possible misconception is that, after one door is eliminated and two remain, many people will feel that as an "either/or" choice, each door has probability of success 1/2, whereas in fact the distribution is not uniform. You might test the possibility of this being the problem by posing other, simpler questions about non-uniform probabilities, for instance: play a game where I flip a coin and give it to you if it comes up heads. What's the probability after two flips that you have 0, 1, or 2 coins? If they know the right answer, ask them to explain why, and if not, work it out with them.

(This can become Monty-Hall-like if you change the game once they understand. I flip both in secret and announce that there were "more than zero" heads, without saying exactly how many. Now what are the probabilities? This is very close to one of the branches of the actual Monty Hall problem)

Because of the coins setup where the "doors" are dynamically generated rather than static and the implication that the game is repeatable is more obvious, it may conform better to the intuition. The language a problem is stated in is very important for it being intuitive: people have different opinions about different but isomorphic versions of a single problem. I don't claim that everyone will like coins either.

I would avoid an explanation that rests upon the other person successfully adopting a mathematical mindset. For instance, the "100 doors" version immediately displays the problem but the concept of specializing from a generalization, or solving a related problem, especially one with extreme behavior, is not always natural to people.

• Think I should put "my" method in as an answer of its own? [not really my own...it belongs to whoever explained it to me properly (though not first -- whoever that was did not do a good job;)] – Tutor Jun 15 '14 at 13:33
• True....but it's still a good method when the other party is not too proud to look at things a different way – Tutor Jun 15 '14 at 13:53
• "This question shouldn't be about "all methods to explain Monty Hall" anyway" .....too late :( – Tutor Jun 17 '14 at 2:30
• @Tutor It was timely when it was written :) – Ryan Reich Jun 17 '14 at 4:05
• When I tried to explain the Monty Hall problem to may Grandpa's CNA, she insisted you should always hold because there was a reason you picked it in the first place. I was unable to successfully counter this belief. – hildred Jun 18 '14 at 16:56

For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me.

Rather than talk about probabilities as fractions, I explain it this way:

"If you picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching."

(stop here and make sure they get that. Then...)

"Suppose you do the experiment 100 times*, and always stick with your initial choice. About 33 of those times, you picked the car on the first choice. So if you always stick, you'll end up with the car 33 times out of 100.

However, about 66 of the times, you didn't pick the car on the first choice, but if you switch, you'll switch to the car.

Therefore if you always switch, you'll get the car about 66 times out of 100, whereas if you always stick, you'll get the car about 33 times.

So it's twice as likely to get the car if you switch, as if you stick.

*or 99 if they're picky

• Very nice. The only thing to spell out is the fact that Monty knows to remove a non-winner, so in effect the '2/3' is packed into that one remaining door. But your explanation is great, and nicely avoids the 100 door extension. – JoeTaxpayer Jun 15 '14 at 13:48
• I have always prefered the 100 door extension and don't understand how it would be difficult to explain in that way. – David G Jun 15 '14 at 17:24
• @JoeTaxpayer: I think the thing which makes this explanation so good is that you don’t have to say anything slippery like “the 2/3 is packed into the one remaining door” to make it rigorous. Monty always picking a non-car door is encapsulated in the step “If you picked the car, you’ll lose it by switching,” etc. — which is a deterministic step. It avoids using conditional probability (which for many people is unintuitive) by breaking the argument into a conditional-but-deterministic part, plus a non-conditional probability part. – Peter LeFanu Lumsdaine Jun 15 '14 at 21:16
• @DavidG The reason I don't like the 100 door version is because the intuition it gives you- for me at least- is not based on Monty knowing which door the car is behind. If Monty picked randomly (with the possibility of him unknowingly opening the door with the car) then switching would bring no benefits. So any explanation that gives you the right intuition in the case where he knows which door it's behind should also gives you the right intuition in the case where he doesn't. Otherwise you're just replacing one misleading intuition with another. – Ben Aaronson Jun 16 '14 at 18:52
• Both of the previous two comments perfectly demonstrate my objection that to someone who doesn't know what's going on, a generalization of a tricky problem may not convincingly illustrate the original problem at all. You have to know how it generalizes, and which features are relevant. – Ryan Reich Jun 17 '14 at 4:08

There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't waste your time on repeating this experience, unless you should find pleasure in doing so.

It should also be noted that for experts in probability theory there is a lot of nitpicking to be done about the exact hypotheses that need to be formulated in order to get a completely well defined probabilistic problem with the purported solution as correct answer (the Wikipedia page on the subject illustrates this). So if the resistance you get is of the type: "it is not so clear cut, it might depend", then very possibly they could actually be right, unless you took all the required precautions in explaining the problem.

• I think you raise a good point, and I always felt that some stories on some mathematicans having a hard time to understand it, is actually based on them not getting a complete description (for example glossing over the fact that the host would never open the door with the prize in the first round.) However, perhaps you could add something to make your post more of an answer to the question, like stressing that explaining the initial set-up again can be helpful in some cases. – quid Jun 15 '14 at 9:07
• I upvoted this answer because I've ran into people who blindly assert the 2/3 solution regardless of the game that is being played. It is absolutely imperative to explain that the host always opens one non-winning door regardless of if your choice is correct or not, and there are some other nit-picky things to be considered as well. If you fail to explain this, Monty could play by different rules - for example, only open a door if you chose the right door initially. – Kikanaide Jun 16 '14 at 16:17
• @Kikanaide: The real Monty Hall would sometimes open the player's door immediately (instant win or loss), would sometimes open a door with the prize (instant loss), and would sometimes open an empty door. If the host--unlike the real Monty Hall--always opens an empty door and allows a switch, then switching will be a winning strategy 2/3 of the time, but if the host doesn't always allow the choice, the probability that the original guess was wrong in those cases where the host does allow a switch may be anywhere from 0% to 100%, depending upon the host's strategy. – supercat Jun 17 '14 at 23:13
• @Kikanaide: Suppose one chooses door #1 and the host always picks the lowest available door. Then there will be a 1/3 probability of the host showing door #3 (in which case the prize must be behind #2) or a 2/3 probability of the host choosing door #2 (in which case, the prize would be just as likely to be behind #1 as #3). That wouldn't change the fact that if the prize was uniformly randomly placed, and the host always shows an empty door and allows a switch, the probability of winning by switching if one doesn't know the hosts reasons for picking the door he did would be 2/3. – supercat Jun 18 '14 at 19:32
• Simul-posting fail: @supercat, sorry I deleted my initial post (and then my next one). My claim is that while the odds of that strategy succeeding are 2/3 in the expectation, the odds are no longer 2/3 once a door is opened. And I don't claim that there exists a bias by which switching becomes unfavorable, or that alters the 2/3 in the expectation. Simply that without that constraint, the odds in a particular game may not be 2/3. – Kikanaide Jun 18 '14 at 19:32

Perhaps it's not the explanation that's the problem. I suggest you have them explain to you their understanding of the problem. Listening to their justification might reveal why your explanation is not gaining traction.

Even in the cases where people are saying they now agree with you, you don't necessarily know that they understand the problem. It's possible that all you did was get them to agree with you. People often just agree with another person who seems to have a better grasp of math.

However, if they explain it to you, you'll have a much better idea of their understanding. No matter whether they agreed or disagreed with your conclusions.

• This is usually very helpful. Alas, some people just don't know why they think what they do. – Ryan Reich Jun 15 '14 at 13:34
• That's why every chance to help them reflect on that fact is a chance worth taking. Otherwise they may never realize the value of their own efforts to explain and justify. If we can't establish justification as a basic mathematical practice, we're not really teaching mathematics. – JPBurke Jun 15 '14 at 16:53
• Usually if they don't believe that it's better to switch, they see the problem as a choice between two doors. The difficulty is then to show them that it's not a choice between two equivalent options, in the sense that a coin flip is such a choice. – ChrisA Jun 18 '14 at 6:22
• @ChrisA Yes, essentially, the disagreement is about equivalence vs. non-equivalence. I think what we, as math educators, have to help students do, though, is see that they have to go beyond saying that the choice is equivalent for their explanation to be considered a justification. How do they show you their basis for believing it is equivalent? If it is similar to a coin flip, why is it similar? Whether or not they can explain it, it certainly helps to establish what counts as justification, and the norm of requiring justification. – JPBurke Jun 18 '14 at 14:21

I've had best luck by simplifying the problem to asking whether you want to pick one door or two doors. Everyone understands they'd be better off picking two doors. At that point I tell them to just ignore the fact that Monty showed them what was behind one of the doors they didn't pick. They're still picking either one door or two doors.

Edit: The more sophisticated, but still confused listened will object that this ignores the additional information gained from Monty opening one of the doors. But that's a red herring. There's always a losing door in the two you didn't pick, so Monty is not revealing any new information.

• Perhaps I am one of the confused, but how is switching equivalent to picking two doors? Yes, you make two choices, but you don't benefit from actually having your fingers in both jars, so to speak. You still only get the car if it's in your final choice, however much more probable that turns out to be. – Ryan Reich Jun 17 '14 at 4:44
• @RyanReich Assuming you switch: you will always switch to the door that has the car (if either of the two unchosen doors has the car) because the other one was eliminated. So you are effectively choosing both (both of the initially unchosen doors) – Richard Tingle Jun 17 '14 at 12:22
• Yeah, but the other door doesn't always have the car. – Ryan Reich Jun 17 '14 at 13:04
• @RyanReich if you imagine the doors as sets, two empty one containing a car, then with the switching strategy you get the union ("both") of the two doors you did not pick originally. – quid Jun 17 '14 at 21:32
• @quid Oh, that makes sense. I was misled by the fact that, in a way, neither of the two doors you pick is actually your choice: Monty chooses one, and you are stuck with the other (if you desire). Nonetheless, two are better than one, as long as Monty doesn't conspire against you. – Ryan Reich Jun 17 '14 at 22:13

I found the most helpful way to think about the problem is to expand it to a larger number of doors. For example, if you have them select from 100 doors, where 99 are losers and 1 is a winner. Then after the initial selection is made, eliminate 98 doors.

The crux of this explanation is from one of Polya's heuristics, as explained below.

The initial answer remains above, as I feel it is a quick answer to the question, thought it might not satisfy everyone. I answered almost immediately after the question was posed, and in the time since this edit, many more thorough answers exist on the board that should satisfy such readers.

I do wish to provide an answer to the comment by @Tutor who asked "what Polya's heuristics are". George Polya is a well-renowned 20th century mathematician whose seminal work is How to Solve it. In it, he discusses his heuristics (techniques) for solving (perhaps difficult) problems. Under the principle of "Devise a Plan" he discusses the idea of solving a different problem if you are having difficulty solving the one posed. Sub-heuristics of this are: (1) solve a simpler problem, (2) solve a special problem, (3) solve a more general problem. The statement of a new problem with 100 doors could fall under any one of these categories.

• I have never understood how this is even supposed to begin to be helpful. Nothing about it triggers any sort of intuition for me; you might as well be talking about the color of Monty's tie for all the relevance I've been able to see. What is this supposed to trigger in the learner's head? – user2357112 Jun 15 '14 at 8:07
• @user2357112: This argument does not intend to transmit any intuition. It specifically targets at destroying the blind conviction that two possibilities that are equally possible are therefore equally likely; one this fallacy is abandoned, maybe the door to a proper analysis will be opened (and no goat will be found there ;-). The destruction of conviction is intended to happen by making the conclusion drawn from the fallacy more "evidently absurd". I have no idea about its effectiveness, but this is clearly the aim. – Marc van Leeuwen Jun 15 '14 at 12:06
• Andrew: maybe edit in part of Marc's comment as your answer to clarify.....also IDK what Polya's heuristics are....maybe add a link to a Wikipedia page or other explanation? – Tutor Jun 15 '14 at 13:18
• This is precisely how I first made intuitive sense of this problem. The essential realization that worked for me is how absurdly unlikely it would be to choose the correct door from among 100 doors. I realized that "switching" was the same as being able to flip the probability (instead of the number of doors being against you, now they were working in your favor). Essentially: let's say the door is chosen for you at random. Would you rather have that door, or the 99 other doors? I'm not saying this is a good explanation for others, but it was my own thought process. – JPBurke Jun 15 '14 at 17:00
• The reason why the 100 doors version is not necessarily helpful is because of where people's idea of 50% chance is coming from. If it's coming from the fact that there are two options for what is behind the doors, then it may help. But if it's coming from the fact that after all doors are open, you have two doors to choose, then it is unlikely to help. I find it interesting that people stick to their own preferred explanation despite the obvious evidence that it doesn't work for everyone. – DavidButlerUofA Jul 23 '14 at 19:44

I never understood it until my Maths lecturer explained it to me. Unfortunately I can't remember the exact way he explained it, but I'll try my best to remember. I really like your own explanation, so there is a chance that the following won't work. In that case, you may have to resort to empirical experiment. Another observation from my own experience is that sometimes I get too "mathsy" too quickly, and people don't follow my reasoning. Perhaps you need just need to slow down a bit!

Try this explanation. Get three cards to represent the doors. Use an ace as the winning door. Now, take two of the cards, and give one to your friend. There is a 2:3 chance that you have the ace, because you have two of the three cards, and there is a 1:3 chance that your friend has the ace, because he has one of the three cards. Now, peek under your cards, and discard one. Ask your friend if they want to swap for the card that you kept.

This is essentially identical to what happens in the problem, but it feels different because you are choosing two of the cards. You can highlight that you had a 2:3 chance of having the ace, because you had 2 of the cards. This may appeal to their intuition.

• By bringing physical aids such as cards into it, you add a visual component. This increases the chance of visually confirming that, by switching, you are in effect switching to both of the non-chosen cards. And we are guaranteed that (1) at least one of those is a loser and (2) will be discarded by Monte. Now, ask if they'd be willing to switch their one card for the two remaining cards before Monte looks at and throws away a loser. – user2338816 Jun 17 '14 at 5:49

The best sure-fire method for teaching this to people who don't want to learn it is to set up a Monty Hall style game for small stakes of real money. When they start losing 2/3 of the time, they will become more receptive to your explanation.

• This seems somewhat related to the set up an experiment/simulation already mentioned. But then to add a more noticeable consequence for some could actually be important. So, it seems like an actual variation of the answer. – quid Jun 15 '14 at 22:52
• Yeah, the money is important. It changes some people's perceptions. – Walter Mitty Jun 16 '14 at 12:14
• There's an online demo at onlinestatbook.com/2/probability/monty_hall_demo.html which helpfully keeps a tally of your successes as well. – Superbest Jun 17 '14 at 7:55
• This works pretty well if you're explaining it to a programmer who can quickly simulate the gambling - and losing - of large sums of money very quickly! – bye Jun 17 '14 at 12:38
• The people who can't be persuaded by logic aren't going to be persuaded by a simulation either. If the logician can lie to them, why can't the author of a simulation? – Walter Mitty Jun 18 '14 at 11:05

While your explanation is correct, and easily understood by people like us, it's a bit too terse (a quality we like) for a lay person to understand. You can simply expand on it. And most importantly ask the audience questions along the way. Every explanatory sentence should have a question that goes a long with it. The key is for you to identify where the hangup is. I like to think about the problem in terms of the chosen door rather than the switching door. So this is how I might go about asking the questions.

1. Ask them before removing a door "what is the probability that the door they chose has the car?"

2. Then say, "I remove a door. Since I am not evil, I am not going to remove the door with the car." Then ask "Why does the probability that the car is behind the door you initially chose not change?" If they don't get this right, work out the cases, as others have mentioned (continuing to ask questions along the way).

3. Then say, "Since there is 1/3 probability that the car is behind your door, what must be the probability the car is behind the other door?"

Explanation at each stage may be required, but this socratic method seems to work well. I deployed something similar to a bunch of small children at the NY county fair. By making the argument composed only of leading questions, its a way of making sure they are following at every step. Of course you can ask them different questions. The key is to make it socratic.

• I think things can be made much clearer if one makes the probability of the host showing an empty door and allowing the player to switch contingent upon whether the player initially guessed correctly (the real Monty Hall would sometimes show the door with a car without offering the player a chance to switch). If the probability of being allowed to switch is affected by the correctness of the original choice, then the fact that one is offered a car should affect one's estimated probability if the original choice having been correct. Note that a "neutral" host would pick a door at random... – supercat Jun 16 '14 at 21:15
• ...and only allow the player to switch if the randomly-picked door didn't hold a car. In that case, a player whose initial choice was right would always get a chance to switch, while one whose initial choice was wrong would only be offered a chance to switch half the time. Switching under such circumstances would be a 50-50- proposition. An evil host would always show the car if it wasn't behind the player's door, and a massively benevolent one would only show it if it was behind the player's door. To always show an empty door and allow switching is mildly benevolent, not neutral. – supercat Jun 16 '14 at 21:19

I find that most people who THINK they understand the Monty Hall Problem, actually don't. For about 5 years I was one of them, until a further insight made me understand it better. More of that at the end.

However, I came up with a super-exaggerated version that seems to give people pause for thought at least.

In the UK, our lottery has a probability of 14,000,00:1 for the jackpot. Suppose I buy all 14,000,000 tickets one Saturday and let you choose 1 ticket with your favourite numbers, the rest I put in my shed. I lock you in a room until the draw with no access to the lottery result and I sit in my shed and watch the lottery draw. After the draw, I walk out of my shed with 1 ticket in my hand and set light to the entire shed and the rest of the tickets. I unlock the door of your room and announce, “Good news! One of us has the winning ticket! Would you like to swap with me?” Why would you think that your ticket has suddenly become a 50/50 chance of being the winner? If you believe that, imagine we did the very same thing NEXT week and the week after. Why has your lottery ticket suddenly acquired a 50/50 chance of winning every week – yet never wins it?

The crucial thing I alluded to at the top is that Monty HAS TO KNOW where the car is before any of the normal interpretations become valid. The “100 doors” scenario illustrates the point very well here. You pick door one and Monty opens door 2, then door 3; all the way to door 36. He bypasses door 37 and moves to 38, 39; all the way to 100. He then asks if you want to swap door one for door 37. Clearly the car is behind door 37 (well, 99 times out of a 100).

However, imagine you watch the program for 3 months before you are a participant and every week Monty keeps opening doors to find that the car is behind: door 17, door 81, door 53 – each week he keeps stumbling upon the car BEFORE the point he would give the contestant the chance to swap. When you go on the show, you choose door 1, then Monty proceeds to open each door from 2 to 99 and does not find the car. When he offers to swap door 100 with you, there really is a 50/50 chance that you have the car already and there is no advantage in swapping.

• Interesting....except in your last paragraph, I think that there still is 99:1 chance of winning by switching.....since there are two doors left, the first door you chose is either the car or isn't; and door 100 [in your last paragraph] MUST be the other....since in your first choosing, you had 99% chance losing, switching now is 99% chance winning – Tutor Jun 15 '14 at 20:27
• The real world kinda intrudes on the lottery example. Unless you're a game show host, you wouldn't want to give up a winning ticket. :) So if you come to me after, wanting to swap tickets with me, and you already know the outcome...heh. Unless i know you're so rich that that much money means nothing to you, i'd suspect i have the winning ticket, and wouldn't swap. – cHao Jun 15 '14 at 20:31
• @Tutor the last paragraph has to do with how he framed the problem. In this case the host opens doors in order until he reaches the car. If he reaches the last door its a 50:50 chance. This is not how the host behaves in the monty hall formulation (in an n door monty hall formulation, he'd open all the doors in one shot, seemingly at random - so that the position of the door does not give added information), its an illustrative example. – WetlabStudent Jun 15 '14 at 21:11
• @Tutor: Saying "Door A is the car" is equivalent to saying "Every door but A is a goat". If Monty knows where the car is, and isn't allowed to reveal it, then all 98 of the doors he opens must be goats, and he has a 99% chance of ending up with the car. (The other 1% is yours.) On the other hand, if he doesn't know, then by eliminating all the other doors, he's effectively choosing one at random -- just like you did. His chances at that point are as good as yours. – cHao Jun 15 '14 at 21:24
• +1 for explicitly confronting the 50/50 misconception with an intuitively plausible scenario. As you say, it may not convince them, but it should make them think about what they believe, which is of course where they are wrong. – Ryan Reich Jun 17 '14 at 4:49

Speaking from my own experience, a stage I had to go through before understanding any explanations of the logic and math involved was indignation at being baffled by a cheap fairground trick that probably goes back centuries. Also the eventual recognition that what put me, the punter, at a disadvantage to the operator was the fact that s/he is working to a strategy and is therefore prepared to lose a game every so often (1 out of 3 in this case), whereas I'm limiting myself to only thinking I've got to win the present game each time.

Another motivational prepping step I'd recommend is a highly entertaining presentation (available on YouTube) by the great tele-mathematician Marcus du Sautoy playing the game with someone who sportingly agrees to be the fall guy who never switches and so loses heavily. This brings out the compelling practicalities.

A couple of points about the explanation stage. A clear diagram with your three doors and a list of the three possibilities of what's behind them: Car Goat Goat; Goat Car Goat; Goat Goat Car. If my first choice of door happens to be the correct one, and I switch, then sure as hell I'm going to lose on that occasion. Don't mince words about that, but use it to show how it means that if instead my first choice happens to be the wrong one, which is going to be 2 times in 3, then switching will reverse those odds into my favour.

(I agree with others that I've never understood how considering lots and lots of doors is helpful)

• And see if this comment helps you to understand 100 doors [it sure helped me;)] – Tutor Jun 15 '14 at 13:13
• That's fine. I just watched it again, and its great strength is it vividly dramatizes the consequences of thinking it doesn't matter if you switch or not. But in my opinion the downside is that to organize the actual explanation round this fallacy complicates things, which is why the video isn't so effective at this stage. Your approach starting with the consequences of wrong and right picks is the clearer and more elegant. But maybe both are needed when dealing with what's essentially a trick. – Wanderlust Jun 16 '14 at 9:33
• @Tutor Sure, but if your explanation requires further explanation in order to understand why it helps, then is it really useful as an explanation? – DavidButlerUofA Jul 23 '14 at 19:41

Here's my take on explaining it. (And it usually involves a drawing for me :-) )

Assuming the host opens a door the odds of making the right choice without switching the door is 1/3 because it was made before having the knowledge the host provided.

Once you decide to make the switch here is a breakdown of the probability of losing VS winning:

• this is very nice because it actually is tersely clear about the host's nature. I think many people who don't understand the problem actually do understand probability. The problem for many is merely the deliberate muddying of the waters as the question is asked. I have similar grumblings about probabilities of children gender. Like, "what is the probability you have 3 girls in a row?" Verses, "you have two girls, what is the probability you have a third girl?" Usually, when people comment I have three girls, wow, what are the odds? They're not talking about the conditional hypothetical. – James S. Cook Jun 17 '14 at 14:05
• they're talking about traveling back in time and asking, if I have three children, what are the odds they're all girls? That is what makes having 3 girls in a row interesting. – James S. Cook Jun 17 '14 at 14:07

The big problem with the "Monty Hall" problem is that there are many problems that sound superficially the same, but have different solutions. The terms of the game have to be stated very precisely.

As an example, Marily vos Savant's statement of the problem as it is quoted in the Wikipedia article is imprecise. It doesn't state whether the game host must or may open another door, whether he must open a door with a goat or whether he is allowed to open a door with the car, and if he has some freedom of choice, whether he is trying to help you or not. Or whether he is even allowed to "cheat" and swap the contents of the two unopened doors sometimes.

So you listened to the explanation why changing doors is so good, but they just changed the rules - after you choose the first door, if you chose a goat the game host will open the door and say "tough luck, you lost". If you chose the door with a car, the game host will open another door with a goat, so if you swap, you lose. So you see that without stating the rules precisely, you can't give the correct answer.

The "100 doors" explanation doesn't help. So you are the lucky guy that by a one in hundred chance picked the car, and you are shown 98 doors with goats. If you swap, you lose. On the other hand, people who picked a goat will be shown fewer than 98 doors (once in a while, someone will be shown 98 doors, or the game would be given away).

• As commented elsewhere I agree that imprecission of the statement can be an issue. But I think the 100 doors explanation in fact can help, as it has as a side effect that the terms are made more precise. – quid Jun 15 '14 at 17:58
• Still, the distinction between "may open" and "must open" is one that has to be kept clear in the definition. – Walter Mitty Jun 17 '14 at 10:52

The best way I came up with is to draw all 9 possible outcomes.
Prize is in A, you pick A;
Prize is in B, you pick A;
Prize is in C, you pick A;
Prize is in A, you pick B;
and so on.
Each outcome is equally probable and 6 of those will make you win.
I don't think it get's any easier than this.

• What if the person doesn't understand probability? – Ryan Reich Jun 17 '14 at 4:52

The problem is most of the time something in the line of that a switch isn't believed to be helpful, because nothing has changed in the eyes of the person you are explaining the problem to: they consider the chances to be the same as before.

Explaining that something has changed can be the best step to take first: make sure they understand that they are getting more information about the system.

Ofcourse, then you go on, that because of this information, the chances have been changed. Then you can explain that the information is different based on the case you were right (1/3, no real new information) and the case you were wrong (2/3, new information is he tells you were the winner is).

Switching turns a winning guess into a losing guess, and a losing guess into a winning one. Since your guess is twice as likely to be losing as it is to be winning, you do better to switch.

• @Tutor I had a hard time with this problem. This is the encapsulation of the argument that convinced me. Your mileage may vary. I suggest you try it. – Thumbnail Jun 16 '14 at 5:04
• I know -- that's pretty much how I got it also, and how I still think of the answer (see question) ....however, I found myself in a situation where that did not work.... – Tutor Jun 18 '14 at 4:31

Once I set up a bet, using a bunch of (future) lotto drawing. Lotto in Italy uses 90 numbers, and traditionally those from 1 to 30 are marked 1, those from 31 to 60 are marked 2, nd those from 61 to 90 are marked X. I would have taken the role of Monty Hall, and I explained in advance the algorithm which I will use to show a result (among 1, 2, X) which did not appear.

Exposing the algorithm makes people think that Monty Hall cannot usually choose a random door (or a random value, in my example). At least for me, in this way is easier to understand that for the opponent too the choices are not casual.

It helps to get rid of the standard trappings of the problem. The problem beneath is mathematically the same, but makes it clearer why you ought switch.

You are a superhero patrolling a crowded train station. A male stranger comes up to you, and asks you to pick a person at random, out of a crowd of thousands. We'll call your pick person A.

This stranger then tells you that he's part of a math-themed supervillain militant movement. He explains that one of the people in the crowd is their agent, who has a bomb that will blow up the city.

Seeing the worry in your eyes---and a total lack of mathematical thinking given the crisis—he says that they'll even up the odds a bit: he'll narrow all but two of the people in the crowd who may be carrying the bomb to person: A (the person you picked at random without even knowing what you were doing), and B (their agent). He guarantees that A or B has the bomb, which'll detonate in a few seconds.

In this case, who would you think has a better chance of carrying the bomb: your random pick (A) or his pick (B)? If you only had time to disarm one of them, would you go for A or B?

This ought clarify why you “switch” rather than just, say, flip a coin. The odds that the bomb is on A, are a truly random chance from the original cast of thousands. The odds that B has the bomb are obviously higher, since B MUST have the bomb if your original choice was wrong, and your original choice only had a $$\dfrac{1}{thousands}$$ chance of being correct.

Short and sweet summation of the method in the question:

It's more likely that your first choice will be wrong than it will be right. Therefore, if you are given the opportunity to switch, it's more likely that switching will be right than it will be wrong.

• This is literally the same as the method explained in the question. – Ryan Reich Jun 17 '14 at 4:45
• That's absolutely incorrect. It is also more likely that your second choice will be wrong than it will be right, so by the exact same (wrong) argument it is more likely that staying with the first decision will be right than it will be wrong. Any explanation needs to take into account the additional information. – gnasher729 Jun 18 '14 at 17:30
• @RyanReich Yeah, I was just trying to reword the original method so it was possibly more approachable. It helped me to think about it in such terms. – SFlagg Jul 8 '14 at 20:22

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (p.190ff), explain the paradox in a way I hadn't seen before.

Pearl imagines changing the rules to "Let's Fake a Deal," where "Monty opens one of the two doors you didn't choose, but his choice is completely at random." Of course he could open the door containing the car/prize, ruining the game. But his point is that, with this random choice, there is no longer any advantage to switching. If Monty opens a door containing a goat, there is a $$1$$-in-$$2$$ probability of winning either by sticking or by switching.

That Monty's door-opening is not random, but rather constrained by Monty's need to avoid revealing the car, now makes it more clear that significant information is conveyed by his goat door-opening.

He assumes you open Door $$1$$, and then:

Let's Make a Deal, vs. Let's Fake a Deal.

Originally, the chances are 1/3 that the car is behind Door 1, 1/3 that the car is behind Door 2, and 1/3 that the car is behind Door three. If you choose Door 1, your chances of winning are now 1/3. More to the point, the chances are 2/3 that the car is behind Doors 2-and-3 together. This fact will hold true for the rest of the problem.

Monte Hall opens Door 3 and shows a goat, meaning that there are ZERO chances of the car behind Door 3. But the chances are still 2/3 that the car is behind Doors 2-and-3 together. What's left of this "together?" Door 2. ALL of these chances are now behind Door 2, which "inherited" Door 3's chances. So the increase in your chances of winning by "switching," to Door 2, are 2/3 to 1/3, or 2 to 1.

Another example: Suppose there was an election with Candidates A, B, and C, each with 10 votes. Suppose candidate C (Door 3) drops out and says, don't vote for me, vote for B over A (and his supporters follow him faithfully so he gets zero votes). What happens next? Candidate B gets candidate C's ten votes plus his own ten for 20, versus ten votes for candidate A.

• I know it's a lost cause by now, but this question, on this forum, really can't be a repository for everyone's favorite explanation of this problem from the mathematical point of view. It's well known how to solve it; the problem is how to teach it. Just presenting an allegedly correct computation will not convince anyone who doesn't understand all the things that are confusing about this puzzle. – Ryan Reich Jun 18 '14 at 0:49
• I don't want to criticize your solution or you. But I'm not convinced by your answer that its style is good for teaching others, or that it consciously addresses something in particular that needs teaching. It's really just a presentation of your favorite explanation, as-is. – Ryan Reich Jun 18 '14 at 1:02
• @RyanReich: I am a historian "dabbling" in math. As such, I am a better "proxy" for a layman than the average mathematician. The idea of a "windfall" is one that is probably appealing to a layman. It's actually a simple, layman construct: You and I together have 100 dollars. I have 0 dollars. How much money do you have? 100 dollars." I learned to use such explanations working as a math tutor in high school, – Tom Au Jun 18 '14 at 1:15

TL;DR There are still 3 doors even though one of them has been opened.

Fact 1. After your initial choice there is a 2/3 chance that the prize is behind one of the other two doors. Your chosen door has a 1/3 chance.

Fact 2. When the empty door is shown to you, there is still a 2/3 chance that the prize is behind one of those two doors.

Fact 3. Except now you know that one of those two doors is empty, so you choose the door that was not shown. That one door has a 2/3 chance that the prize is behind it.

Conclusion: The unknown door has a 2/3 chance of having the prize, but your chosen door only has a 1/3 chance. So always switch.

The mistake that most people make is thinking that it is a now a two door problem and therefore has a 50-50 chance. It is still a 3 door problem and can be easily solved with basic logic.

• If you think it can be "easily solved" and involves "basic logic" then you are insulting your students. Smart people still have a hard time with this problem, and acknowledging this is the first step to successfully educating them. – Chris Cunningham Jun 16 '14 at 2:15

## protected by quid♦Jun 16 '14 at 8:34

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).


text: The equation of a curve is y =2x^2 + 4x +5. Determine the coordinates of the turning point of the curve


text:
Two columns overlap in table with multirow macro

I am trying to reproduce the diagram below by using the bigdelim and multirow packages and I had problems with the first brace which contains $\mathbb{C}$ Complejos because overlaps the second with $\mathbb{R}$ Reales. I have tried to fix it but I cannot put the columns in the correct position.

This is my attempt

\documentclass{minimal}
\usepackage{multirow,bigdelim}
\usepackage{amsmath, amssymb}

\begin{document}

\begin{tabular}{llllll}
\ldelim\{{9}{3mm}[$\mathbb{C}$ Complejos] & \ldelim\{{8}{3mm}[$\mathbb{R}$ Reales] & \ldelim\{{7}{3mm}[$\mathbb{Q}$ Racionales] & \ldelim\{{6}{3mm}[$\mathbb{Z}$ Enteros] & \ldelim\{{2}{3mm}[$\mathbb{N}$ Naturales] & $\mathbb{P}$ Primos \\
& & & & & Compuestos \\
& & & & & \\
& & & & $\{0\}$ & \\
& & & & & \\
& & & & $\mathbb{Z}^-$ Enteros negativos & \\
& & & Fraccionarios & & \\
& & $\mathbb{I}$ Irracionales & & & \\
& Imaginarios & & & & \\
\end{tabular}

\end{document}

And this is what I get. As you can see, the first two columns overlap and additionally there is an extra space in the last column. What changes in the code are needed?

\documentclass{minimal}
\usepackage{multirow,bigdelim}

\begin{document}
\begin{tabular}{p{24mm}l p{24mm}l l@{\hspace{-7ex}}l}% changed
\ldelim\{{9}{3mm}[$\mathbb{C}$ Complejos]
& \ldelim\{{8}{3mm}[$\mathbb{R}$ Reales]
& \ldelim\{{7}{3mm}[$\mathbb{Q}$ Racionales]
& \ldelim\{{6}{3mm}[$\mathbb{Z}$ Enteros]
& \ldelim\{{2}{3mm}[$\mathbb{N}$ Naturales]
& $\mathbb{P}$ Primos \\
&   &   &   &   & Compuestos \\
&   &   &   &   & \\
&   &   &   & $\{0\}$   & \\
&   &   &   &   & \\
&   &   &   & $\mathbb{Z}^-$ Enteros negativos  &   \\
&   &   & Fraccionarios             &   & \\
&   & $\mathbb{I}$ Irracionales     &   &   &   \\
&   Imaginarios   &   & &   &    \\
\end{tabular}
\end{document}

as tikz picture (edited version):

\documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{calc,
decorations.pathmorphing,%
calligraphy,% had to be after decorations.pathreplacing
positioning,
}
\usepackage{amssymb,bbm}
\begin{document}
\begin{tikzpicture}[
node distance=0mm and 3mm,
every node/.style = {minimum height=4mm, text depth=0.3ex, inner sep=1pt, align=left},
BC/.style = {decorate,
decoration={calligraphic brace, amplitude=6pt,
mirror},
very thick,
pen colour={red}
},
]
\node (n1)                      {$\mathbb{C}$ Complejos};
\node (n2)  [above right=of n1] {$\mathbb{R}$ Reales};
\node (n3)  [above right=of n2] {$\mathbb{Q}$ Racionales};
\node (n4)  [above right=of n3] {$\mathbb{Z}$ Enteros};
\node (n5)  [above right=of n4] {$\mathbb{N}$ Naturales};
\node (n6)  [right=of n5]       {$\mathbb{P}$ Primos \\
Compuestos};
\path   let \p1 = (n1.east),
\p2 = (n6.north west),
\n1 = {veclen(\y2-\y1,\x2-\x1)} in
coordinate[below=\y2-\y1 of n1.east -| n2.west] (c1);
\node (n7)  [above right=of c1 -| n1.east]
{Imaginarios};
\node (n8)  [above right=4mm and 3mm of n7.north -| n2.east]
{$\mathbb{I}$ Irracionales};
\node (n9)  [above right=4mm and 3mm of n8.north -| n3.east]
{Fraccionarios};
\node (n10) [right=of n4]       {$\{0\}$};
\node (n11) [right=of n3 -| n4.east]
{$\mathbb{Z}^-$ Enteros negativos};
%
\draw[BC]   (n6.north -| n2.west) -- (c1);
\draw[BC]   (n6.north -| n3.west) -- (n8.south west);
\draw[BC]   (n6.north -| n4.west) -- (n9.south west);
\draw[BC]   (n6.north -| n5.west) -- (n11.south west);
\draw[BC]   (n6.north west) -- (n6.south west);
\end{tikzpicture}
\end{document}

• Thanks @Zarko! Your answer is flawless. Your approach with TikZ is awesome. I will apply that one. If I could give you more reputation points, I would give you them! Simply excellent! – Cragfelt Aug 27 '17 at 5:44
• And thank you for adding the \usepackage{amssymb}. I completely forgot! – Cragfelt Aug 27 '17 at 5:53
• is it terribly difficult to close up the vertical space in the tikz version? (i find it harder to grasp the meaning with all that space.) even the purely tabular version is more spaced-out in the next-to-last column than really necessary, causing lengthening of the preceding columns. (otherwise very nice!) – barbara beeton Aug 27 '17 at 12:45
• it can be more vertically tied, for this you need to change node distance=1mm and 2mm for example to node distance=0mm and 2mm, reduce nedo height to minimal height=4mm and manualy tweak positions of nodes n7, n8 and n9. latter i will try to edit my answer with this new setings. – Zarko Aug 27 '17 at 12:53
• @Cragfelt, see edited answer, for further reduce vertical size of image is possible only with asymmetrical curly braces – Zarko Aug 27 '17 at 13:07

Lately, I found this easy solution with cases environment

\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}

\begin{displaymath}
\mathbb{C}\text{ Complejos}
\begin{cases}
\mathbb{R}\text{ Reales}
\begin{cases}
\mathbb{Q}\text{ Racionales}
\begin{cases}
\mathbb{Z}\text{ Enteros}
\begin{cases}
\mathbb{N}\text{ Naturales}
\begin{cases}
\mathbb{P}\text{ Primos} \\
\text{Compuestos}
\end{cases} \\
\{0\} \\[2ex]
\mathbb{Z}^-\text{ Enteros Negativos}
\end{cases} \\
\text{Fraccionarios}
\end{cases} \\
\text{Irracionales}
\end{cases} \\
\text{Imaginarios}
\end{cases}
\end{displaymath}

\end{document}

Define a suitable environment and then follow the stream.

\documentclass[a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[spanish,es-noquoting]{babel}

\usepackage{amssymb}

\newenvironment{textbrace}
{$\renewcommand{\arraystretch}{1.5}\left\lbrace\begin{tabular}{@{}l@{}}} {\end{tabular}\right.$}

\begin{document}

\begin{table}
\centering\small

$\mathbb{C}$ Complejos
\begin{textbrace}
$\mathbb{R}$ Reales
\begin{textbrace}
$\mathbb{Q}$ Racionales
\begin{textbrace}
$\mathbb{Z}$ Enteros
\begin{textbrace}
$\mathbb{N}$ Naturales
\begin{textbrace}
Primos
\\
Compuestos
\end{textbrace}
\\
$\{0\}$ Cero
\\
Enteros negativos
\end{textbrace}
\\
Fraccionarios
\end{textbrace}
\\
$\mathbb{I}$ Irracionales
\end{textbrace}
\\
Imaginarios
\end{textbrace}

\caption{Jerarquía de números}
\end{table}

\end{document}

• (+1) For the elengant custom environment and for the caption: Jerarquía de Números. Thank you. – Cragfelt Dec 16 '17 at 23:02
• @Cragfelt The translation was provided by the Web. ;-) – egreg Dec 16 '17 at 23:03
• Nothing to be ashamed about. I do that every time. Grazie mille. – Cragfelt Dec 16 '17 at 23:19


text:
+1 vote

The timer logic introduced to the postback mechanism in SKAdNetwork 2.0 is convoluted and confusing. I'll attempt to explain it below, but this article should be seen as necessary background in understanding the SKAdNetwork postback.

Conversion Value

Conversion Value is a parameter that was added to the SKAdNetwork 2.0 postback that is sent from Apple to the ad network that sourced the user to the app. The conversion value is a 6-bit value; a 6-bit value is a sequence of six characters, each of which can only contain either a 0 or 1. A 6-bit value can take 64 different values (2^6 = 64).

The highest possible value for a 6-bit value is 111111. The lowest possible value for a 6-bit value is 000000. To understand how 6-bit values "ascend," or increment, see this article.

Postback Timer

If the developer wants to include a conversion value in its SKAdNetwork postback to the ad network that supplied the user, they should invoke the updateConversionValue() method at app open. Once this method is invoked, a resettable 24-hour timer is started -- when this timer expires, the postback is fired to the ad network.

The 24-hour timer is reset each time the updateConversionValue() method is fired with a 6-bit value that is higher than the 6-bit value currently recorded (or if a 6-bit value is being provided for the first time after app open). From the documentation:

If updateConversionValue() is called with the same 6-bit conversion value as is currently recorded, or with a lower valued 6-bit conversion value than is currently recorded, then the 24-hour timer does not reset. And calls to updateConversionValue() after the postback has been sent do nothing.

Once the 24-hour timer expires, a second, random timer of less than 24 hours starts. This timer is not resettable and is only meant to obfuscate the process such that the app developer won't be able to map a received conversion value at a network with a user who took the action mapped to that conversion value at a specific time.

A few things to note here:

• The SKAdNetwork postback does not include a date parameter, so the ad network receiving these postbacks doesn't know when the user was acquired upon receipt of their postback, and it also doesn't know when the conversion value event was recorded (because of the obfuscation timer);
• Only one postback is ever sent. No subsequent conversion values are sent to the network after the install attribution postback is sent.

The conversion value is an arbitrary mapping that the developer can attach to anything in the app: to a specific event being trigger by the user (eg. complete level 7, finish tutorial, make payment, etc.), or to some imputed LTV of the user at that time, etc. Managing conversion values will be a central component of advertising strategy in iOS14.

by (14.6k points)
edited
Am I right, that it means we can postpone the event window from 24 hours to up to 64 days if we will daily send fake events and use only 111111 as an important event for us? That could help for App developers with long lifetime and let's say mark distinguish bad vs good users by this technique.
updateConversionValue() has to be called from within the client, it can't be called server-side. So assuming the user opened the app every day, then yes, you could delay the event being sent for 64 days.
Just to be clear on the bit math here - 1^0 = 1, so 2^0 + 1^0 would equal 2, not 0. You might want to change it to something like:

0(2^1) + 0(2^0) = 0
0(2^1) + 1(2^0) = 1
1(2^1) + 0(2^0) = 2
1(2^1) + 1(2^0) = 3

The integer value 7 is represented with a 3-bit value as: 111 (1(2^2) + 1(2^1) + 1(2^0)). The integer value 25 is represented with a 5-bit value as: 11001 (1(2^4) + 1(2^3) + 0(2^2) + 0(2^1) + 1(2^0)).

The discussion before the example is a bit off, too. I could try to help rewrite it but I think the discussion of how bits work is somewhat unnecessary outside of the explanation of "You can send back a single number between 0 and 63. You have 24 hours to send another number, but it replaces the previous one and has to be bigger. You only see the very last number sent (after the 24-hour window expires.)"
Whoops, I had quite a few errors in that answer! Thanks for bringing that to my attention.