Engineering telepathy

Brain-to-brain communication isn't a novel concept - we do it all time when we talk and listen to each other. However it would be a notable achievement if we could bypass our normal communication organs (e.g., mouth and ears) and instead directly communicate with each other via our thoughts.

A recent scientific paper describes a cooperative activity between two people involving direct brain-to-brain communication that occurs without using their normal sensory and speech organs. Instead, the first person imagines the physical action required to achieve a certain goal and, via brain signal monitoring, electronic transmission and brain stimulation, a physical action is produced in the second person to achieve the desired goal.

The way the experiment is set up is that two people, in buildings a mile apart, are playing a computer game that requires firing a cannon to shoot down an enemy rocket. The first person can see the game screen but they have no input device to fire the cannon. The second person cannot see the game screen, but they do have an input device to fire the cannon. The goal is for the first person to determine when the cannon needs to be fired, instruct the second person to fire the cannon and consequently destroy the enemy rocket. However the only available communication channel is via direct brain-to-brain communication.

The way the goal is achieved is that the first person imagines moving their hand up or down. Brain sensors on their scalp interpret their brain signals (via electroencephalography, or EEG) and move a cursor on the screen up or down accordingly. When the enemy rocket appears, the person causes the cursor to move up and over a button that represents the instruction to fire the cannon. This instruction is then transmitted via the internet to the second person's computer. When the instruction is received, the second person's brain is stimulated at a particular region (via transcranial magnetic stimulation, or TMS) that causes their hand to move up and then down onto a button which fires the cannon. If all goes according to plan, this action shoots down the enemy rocket.

That's the experiment and it works successfully. While this is a very nice end-to-end demonstration of where the technology is at now, the obvious improvement would be to have the first person just think the instruction to "fire now!" and have the instruction detected and automatically transmitted to the second person. Then the recipient's brain would be stimulated such that it generates the idea of "fire now!" in their minds and, finally, they act by pressing the button.

That would be a truly impressive demonstration of engineering telepathy. As Arthur C. Clarke once proposed, "Any sufficiently advanced technology is indistinguishable from magic."

Are you smarter than a chimp?

I watched a great talk called, "How not to be ignorant about the world" by Hans Rosling. He demonstrates that people, regardless of education level, are often less knowledgeable about the world than they think. Hans jokes in his talk that monkeys randomly selecting the answers to his quiz do better than the people answering them!

To get the idea, try answering the four questions below to see how you go.

1. In 1950 there were fewer than one billion children (aged 0-14) in the world. By 2000 there were almost two billion. How many do UN experts think there will be in 2100?
    
   A. Four billion
   B. Three billion
   C. Two billion
    
2. What percentage of adults in the world today are literate – can read and write?
    
   A. 80%
   B. 60%
   C. 40%
    
3. On average, in the world as a whole today, men aged 25-34 have spent 8 years in school. How many years on average have women in the same age group spent in school?
    
   A. 7 years
   B. 5 years
   C. 3 years
    
4. In the last 20 years the proportion of the world population living in extreme poverty has...
    
   A. Almost doubled
   B. Remained more or less the same
   C. Almost halved

You can find the answers to the quiz at the end of this post along with the percentage of people who selected the correct answer. The questions were part of the Ignorance Survey that was conducted in the US in 2013.

So how did you go? If you didn't do very well then you're in good company with the vast majority of respondents! As you can see by the percentages, the respondents did worse than monkeys randomly choosing the right answers 33% of the time.

One of the issues Hans raised was whether the problem was due to people not reading and listening to the media. It turns out that after surveying the media with these questions, they fared no better than the general population. The problem was that even the media themselves didn't know basic facts about the world!

In the second part of the talk, Han's son Ola provided some tips for beating the chimps. He pointed out that the reason that we often get these questions wrong is due to:

1. Personal bias - we tend to generalize from our own experience which is not representative of the broader population
2. Teachers often teach outdated information based both on what they learnt during their schooling and on outdated text books
3. News bias - sensational and unusual events are more likely to make headlines and these are not representative of world events

These biases lead to the misconceptions that:

1. Everything is getting worse
2. The gap is increasing between rich and poor
3. People need wealth before social development can occur
4. Sharks are dangerous

Ola suggests that people's scores on the world fact survey will immediately and dramatically improve if they generalize in the opposite direction, recognizing instead that:

1. Most things improve
2. Global wealth can be represented as a normal bell curve with most people in the middle
3. Most people are already socially developed before they have wealth
4. Sharks are not actually very dangerous to us since they kill very few people - so recognize that fear exaggerates danger

--
Answers to the quiz (along with the percentage of respondents who selected the correct answer):

1. C (7%)
2. A (22%)
3. A (24%)
4. C (5%)

 

The Blue Eyes Puzzle (Part 2)

This follow-up post gives the solution to the Blue Eyes puzzle described here.

The answer is that the 100 blue-eyed people will leave the island on the 100th day.

The way to understand this answer is to first work through a much simpler version of the problem where there is only one-blue eyed person and one brown-eyed person on the island. Let's first try to solve it before the outsider arrives. The blue-eyed person doesn't see anyone with blue eyes, while the brown-eyed person sees one person with blue eyes. Since the blue-eyed person doesn't know whether there are any blue-eyed people on the island, he doesn't know whether he himself has blue eyes. So he will not leave the island.

However once the outsider informs the people that there is at least one blue-eyed person, the single blue-eyed person immediately knows that it must be him. Thus he leaves the island that night. Conversely, when the outsider makes her statement, the brown-eyed person doesn't yet know the color of his own eyes because he can already see someone with blue eyes. But he knows that if he does have brown eyes, then the other person will deduce their own blue eyes and leave the island that night. Or if he himself has blue eyes, then the other person will similarly not yet know the color of his eyes and will stay on the island. Once he observes the blue-eyed person leave the island that night, he then immediately knows that he himself has brown eyes.

So in this simple case, at least, the outsider's statement conveyed crucial information to the blue-eyed person that he didn't previously know and that made the difference to whether he left the island or not. Also, on observing the blue-eyed person leave the island, the brown-eyed person deduces that he himself has brown eyes.

Now consider the next simplest case where there are only two blue-eyed people and two brown-eyed people. Again, let's try to solve it before the outsider arrives. Each blue-eyed person can see one other blue-eyed person, while the brown-eyed people can see two people with blue eyes. Therefore everyone knows that there is at least one blue-eyed person on the island.

The first blue-eyed person realizes that if he himself has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. However, without the outsider's information, the other blue-eyed person cannot deduce that he has blue eyes and will therefore stay on the island. He similarly realizes that if he has blue eyes, he has no way of knowing this, so will stay on the island.

However when the outsider states that there is at least one blue-eyed person, the first blue-eyed person will now reason differently. He again realizes that if he has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. But he now knows that the other blue-eyed person would deduce that he himself has blue eyes and leave the island on the first night. So the first blue-eyed person now just has to wait until that night to see if the other blue-eyed person leaves the island. If he does, then he can deduce that he himself has brown eyes. If he does not leave, then he can deduce that he himself has blue eyes, and will leave on the second night. The other blue-eyed person reasons in the same way and also leaves on the second night. The brown-eyed person also reasons in a similar way. However, because he can see two blue-eyed people, he will wait until the second night to see what happens. When he observes the blue-eyed people leave the island on the second night, he then immediately knows that he himself has brown eyes.

So the outsider's statement, which only seemed crucial in the one blue-eyed person scenario was also crucial in the two blue-eyed people scenario. That's because it allowed each blue-eyed person to rule out the one-blue-eyed person scenario by observing what happened after the first night. It also allowed each brown-eyed person to confirm the two blue-eyed people scenario after the second night.

The same line of reasoning applies to the three blue-eyed people scenario. Without the outsider's statement, no-one will leave the island. With the outsider's statement the three blue-eyed people can rule out the two blue-eyed people scenario after the second night, deduce that they themselves have blue eyes and leave on the third night.

How do they conclude this? All three blue-eyed people hypothesize that if they themselves have brown eyes, then there are only two blue-eyed people on the island who, in turn, hypothesize that if they themselves have brown eyes, then there is only one blue-eyed person on the island. That hypothesized single blue-eyed person will deduce that he has blue eyes and leave on the first night as a result of the outsider's statement. But if no-one leaves on the first night, then that falsifies the hypothesized two blue-eyed people's hypothesis, so they would leave on the second night. But if no-one leaves on the second night, that falsifies the hypothesis of the three blue-eyed people, so they will all leave on the third night.

Conversely, the brown-eyed people will observe the three blue-eyed people leave on the third night and deduce that they themselves have brown eyes. The same style of reasoning applies to the 100 blue-eyed people scenario, with the blue-eyed people leaving on the 100th night and the remaining islanders deducing that they themselves have brown eyes.

So the outsider's statement did enable everyone to deduce their eye colors. But what new and useful information did it convey? In the single blue-eyed person scenario, it informed the blue-eyed person that someone had blue eyes. In the two blue-eyed people scenario, it informed the two blue-eyed people that the other blue-eyed person knew that someone had blue eyes. In the three blue-eyed people scenario, it informed the three blue-eyed people that the other blue-eyed people knew that the other blue-eyed people knew that someone had blue eyes. And so on.

In other words, the information that at least one person had blue eyes had become common knowledge. This meant that they all knew it, they all knew that they knew it, they all knew that they all knew that they knew it, and so on ad infinitum.

Another way to think about this puzzle is that the blue-eyed people know that they are either in a 99 or a 100 blue-eyed people scenario, but they don't know which one. Whereas the brown-eyed people know that they are either in a 100 or a 101 blue-eyed people scenario, but they also don't know which one. So the blue-eyed people will wait to see if the other blue-eyed people leave the island on the 99th night. If they do not leave, then the blue-eyed people deduce that they are in the 100 blue-eyed people scenario, and leave on the 100th night. The brown-eyed people observe the blue-eyed people leave the island on the 100th night, deduce that they are in the 100 blue-eyed people scenario, and thus know that they themselves have brown eyes. But remember that no-one will leave the island without the outsider's statement since her statement is necessary to falsify or confirm each persons' deeply-nested hypothetical about what the others know.

The key to solving this puzzle is to recognize that it has a recursive structure, observe the pattern that emerges with the simpler scenarios, and then apply that pattern to the more complex 100 blue-eyed people scenario. It's like the pattern that emerges when solving the factorial of four (or 4!). 4! = 4 * 3!, 3! = 3 * 2!, 2! = 2 * 1!, 1! = 1. Once the base case of 1! is solved, and the factorial pattern is understood, it's then easy to solve for higher-numbered factorials.

For some other expositions of the blue eyes puzzle, see xkcd and Terence Tao.

[Added Sep 24]
Bonus question: In a different version of the puzzle, the outsider comes back the day after making her public announcement, calls together all the people on the island, and makes a new public announcement:

"I'm terribly sorry, but I retract my statement from yesterday. I had a migraine that caused me to not see colors correctly so I don't actually know that I saw someone with blue eyes."

What effect (if any) will the outsider's new statement have?

The Blue Eyes Puzzle (Part 1)

I came across a fascinating logic puzzle that goes like this:

On an island, there are 100 people who have blue eyes and 100 people who have brown eyes. No-one on the island knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island that night. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color. Also it is public knowledge that the islanders are perfect logicians - if a conclusion can be deduced then they will immediately do so.

One day, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement:

"I can see someone with blue eyes."

Who leaves the island and on what night do they leave?

If you haven't seen this puzzle before, I encourage you to think about it before continuing.

...

All done?

Is your answer that no-one will leave the island? If so then I'm sorry to say that you are incorrect! Despite the fact that all the islanders can see many blue-eyed people, the outsider's statement does convey new information to the islanders and that information enables them to determine the color of their own eyes.

I will give the actual answer and explanation in a few days time. In the meantime, here is a suggestion to help you find the correct solution.

First try to solve a much simpler version of the puzzle. The simplest version is the scenario where there is one blue-eyed person and one brown-eyed person. Consider what the islanders can deduce before the outsider arrives. Then consider what they can deduce after the outsider makes his statement.

Now solve the next simplest version where there are two blue-eyed people and two brown-eyed people. Again consider what the islanders can deduce both before and after the outsider makes his statement. This will help you notice what the islanders learned from the outsider's statement and how it is necessary for solving the subsequent versions with three blue-eyed people and so on.

[Added Sep 22: The Blue Eyes Puzzle solution]

Visualizing Euler's Identity

Euler's Identity is often considered the most beautiful equation in mathematics because it elegantly combines one each of 0, 1, i, e, π, addition, multiplication and exponentiation. But intuitively, what does it mean?

Answering that requires a brief tour of exponential growth (to understand e) and complex numbers (to understand i and π). My previous post on complex numbers is here.

e, or Euler's number, is a mathematical constant (approximately 2.72) that represents 100% continuous growth when starting at 1. For example, 100% compound interest on $1 for a year would be e in dollars, or $2.72. With continuous growth, the interest is calculated at every instant as compared with yearly (where you would only end up with $2), quarterly ($2.44), monthly ($2.61) or daily ($2.71). The formula for calculating compound interest is:

  (1) compound interest = (1 + 1 / time periods)time periods

And e is the limit that is approached as the number of time periods increase.

e is used to calculate continuous growth for any rate, time period or starting point. So 10% compound interest on $1,000 for 3 years would be $1,000 * e ^{0.1 * 3} = $1,349.86. The formula is:

  (2) final amount = initial amount * e rate * time periods

If we know the initial and final amounts but not the interest rate, then the natural logarithm function is used, as follows:

  (3) rate = ln(final amount / starting amount)

The natural logarithm of a number is the exponent that e is raised to in order to get that same number. For example, 2 = e^0.69, so the natural logarithm of 2 is 0.69. That is, the compound interest rate required to grow from $1 to $2 in 1 year is 69%.

This is where I think things get interesting! Any number can be interpreted as the end result of continuous growth starting from 1, that is:

  (4) number = e ln(number)

This same idea can be applied to numbers with exponents. For example, 2³ can be understood as 1 growing to 2 at a rate of ln(2) (or 69%) and then again to 4 and finally to 8, for a total of 3 growth periods. That is, 2³ = e ^{ln(2) * 3}. Generalizing, we get:

  (5) number time periods = e ln(number) * time periods

In geometric terms, continuous growth can be visualized as a scaling operation on the real number line (i.e., the distance travelled in each growth period is increasingly larger, as with 1, 2, 4, 8, 16, ...) However if a logarithmic scale is used for the number line then the distance travelled in each growth period will be the same (the equidistant points can be marked as 1, e, e², e³, ...)

As with real numbers, imaginary numbers also can be interpreted as the end result of continuous growth starting from 1. However instead of exponential growth along the real number line, imaginary growth follows a linear circular path around the origin on the complex plane. The reason the growth is linear is because the scale is logarithmic with a growth rate of i represented by an angular distance of 1 radian. You can see this linearity when multiplying by i. 1 multiplied by i rotates 1 by π / 2 radians (90 degrees) to i and multiplying by i again rotates a further π / 2 radians to -1.

To summarize, a growth rate of 1 (or 100%) results in growth from 1 to e¹ on the real number line, whereas a growth rate of i results in growth from 1 to eⁱ on the complex number plane - a distance of 1 radian around the unit circle (see the diagram below).

Similarly, a growth rate of 2i will travel 2 radians from 1 to e²ⁱ. A growth rate of π / 2i will travel π / 2 radians to e ^{π / 2 i} or i. A growth rate of π i will travel π radians to e^{π i} or -1. And a growth rate of 2π i will travel all the way around the unit circle to arrive back at 1.

That second-to-last calculation is Euler's Identity: e ^{π i} = -1. It just means that starting at 1 and growing at an imaginary rate of π (via an anti-clockwise rotation), we will end up at -1 on the real number line. A way to remember this is that if you deposit $1 in a bank offering π imaginary interest, you'll end up owing them $1 after 1 year! Fortunately, if you wait for another year, you will get your original money back...

The basic formulas for calculating real growth can easily be applied to imaginary numbers. The key insight is to transform the number to base e and then the growth rate will be expressed in the exponent (as imaginary or real). Some further fun equations to end this post.

  (a) 23i = e ln(2) * 3i = e 0.69 * 3i = 2.08 radians

Start at 1 and grow at a 69% compound rate three times for a distance of 2.08 radians.

  (b) i = e ln(i) = e π / 2i = π / 2 radians

Start at 1 and grow at a rate of π / 2 radians to arrive at i.

  (c) i2 = e ln(i) * 2 = e π / 2i * 2 = e π i = π radians = -1

Euler's Identity derived from the complex number identity i ² = -1.

  (d) ii = e ln(i) * i = e π / 2 * i * i = e π / 2 * -1 = e -π / 2 = 0.21

Start at 1 and grow at a rate of -π / 2 to arrive at 0.21.

  (e) (ii)i = e ln(ii) * i = e ln(e -π / 2) * i = e -π / 2i = -i

Start at 1 and grow at a rate of -π / 2 radians to arrive at -i.

Lunchbox game theory

A practical lesson in game theory: the three-person game.

When helping the kids pack yoghurt for their lunch boxes, I found there was only strawberry yoghurt left in the fridge in the tear-off individual containers. Their preferred yoghurts, mango and vanilla, were finished.

So I asked Liam to grab new yoghurt from the garage fridge. He came back with two large containers (there were no tear-off packs left) which happened to be mango and vanilla. Since we could only open one of them, I asked Liam which one he wanted.

"Mango."

Liam then asked Michelle the same question, who replied, "Vanilla."

I said, "Sorry, we can only open one. So which one will we have?"

Same responses.

"Michelle.", I said, "We can only have one, so how about we have mango this week?"

"No!", she says, "I want vanilla!", as she starts to melt down.

"OK, Liam.", I said, "How about we have vanilla this week, since it seems to be important to Michelle?"

"No!", he says, "I want mango and I said it first!", as he also starts to melt down.

"OK.", I said, "Since you can't agree, I'm going to put them away and you can both have strawberry today."

"Noooooooo!"

OK that was traumatic. Unfortunately, in addition to the yoghurt wars, I see that tomorrow we will only have one bread roll left and one of the kids will have to have sliced bread...

Perspectives on Islam

I watched an excellent Richmond Forum discussion exploring whether Islam is a religion of violence or peace. The first speaker was Ayaan Hirsi Ali, a well-known activist who is critical of Islam. The second speaker was Maajid Nawaz, a former Islamic radical who now promotes democracy in the Muslim world. The final speaker was Imam Feisal Abdul Rauf who leads a mosque in downtown Manhattan and works at improving relations between the Muslim world and the West. A few years ago I read his book, "What's Right with Islam: A New Vision for Muslims and the West".

Ali's view is that Islam is fundamentally a religion of violence and subjugation and that this is the natural interpretation of particular passages in the Quran and hadith literature. She believes that extremists provide a compelling and inspiring case arguing from the Quran and that this is the main reason why they find ready and devout followers. For her, the only viable solution is for an Islamic reformation to occur where such passages are fully repudiated. Without this, a political and authoritarian Islam will continue to threaten the world.

Nawaz makes the case that Islam is not intrinsically a religion of either violence or peace.  Rather, there are competing interpretations that promote political violence and social discrimination on the one hand and liberal and democratic ideals on the other hand. What is important is to support people who are looking to reform the discourse and open up the debate in Muslim countries. He also points out the widespread problem of half-truth narratives. This occurs when religious texts or world events are cherry-picked to create false or misleading perceptions. For example, many Muslims think that Americans are anti-Muslim because certain events are played up in the Muslim world (e.g., the burning of copies of the Quran in Florida or the US president claiming that God told him to invade Iraq). Conversely, many westerners learn about the concepts of jihad or sharia from extremist sources and assume that these are the commonly-held interpretations of Muslims.

Rauf makes a different case again that Islam is fundamentally a religion of peace with its roots in the Abrahamic faith. His view is that Islamic jurisprudence has a venerable tradition, often exemplifying religious tolerance, progressive views for women, and intellectual achievement. He notes that Islamic fundamentalism is a relatively recent phenomenon that is discontinuous with the tradition of Islam, particularly with its insistence on an exclusive Islamic nation state which he sees as an incoherent concept. He also sees the need to change the cultural discourse around Islam. Instead of portraying terrorism and fundamentalism as an Islam versus the West battle, to rather understand it as a religious moderate versus religious extremism battle.

One point that especially struck me during this discussion is the problem of half-truth narratives. It's almost always possible to interpret events and textual sources in ways that support conflicting narratives. The real challenge is to form the most likely conclusions from the data that is available. In this case, the two narratives are "Islam versus the West", which Ali argues for and "Extremists versus Moderates", which Nawaz and Rauf argue for.