Information Visualization Isn’t Easy

Knowing how to best represent a set of quantitative information is not easy. I strongly recommend a quick read of the Wikipedia entry on Misleading Graphs and if you want to read a whole book try The Visual Display of Quantitative Information by Edward R. Tufte. Instead of remphasizing the need for displaying information in the simplest way possible, I’m going to give examples of how not to display information. Sometimes the errors are only small. But those errors will always be notiable, they affect how people interpret the information. First up, WordPress’s graphs

The problem here is that when the view count is 0 the colour is grey – it should just be a lighter shade of yellow. The sudden shift from 1 as yellow to grey as 0 suggests there is a fundamental difference between 1 and 0. There isn’t, as I explained in my my https://emphatious.wordpress.com/2012/10/12/zero/. Wikipedia correctly uses a colour to show a semantic difference. White is used to represent the fact that there is no information available in the following graph.

However, this graph isn’t without its own problems. Can you figure out what the problem is? It’s incredibly difficult to realize the problem as it is a very deep one. Think about colours.

The most obvious problem is that the graph does not use a gradient. groups of GDP are lumped together for no reason. There should be no groups. instead of $800-$1600 being one colours, $900 should be a little more red than $850, $950 a bit more red than $900 and so on. They are essentially rounding figured for no reason. If you had a graph showing $11.53 million profit in Q1 and $12.42 million profit in Q2, would you round them? Rounding them serves no purpose. All it does is reduce accuracy… nothing is improved.

There is a much more subtle assumption made by this graph. It might not be wrong but it isn’t explained. The assumption is that the utility of money increases logarithmically. Look at the groups, they increase logarithmically.

The designers of the graph took it upon themselves to change how the information is shown. Instead of something that’s 10x the size quantitatively being 10x brighter or bigger (10x more intense in some way represented visually) on the graphic, they’ve made it log_10 times brighter. That’s an arbitrary operation done without justification. I do expect that if I were to ask the designer of the graphic why he did this, he would reply with the unforgivable line
“because if it weren’t like that, it’d be hardest to tell the difference between countries with low income. For example, almost all of Africa would be very similar in colour”.
GRAPHICS SHOULD NOT BE USED AS A WAY TO SHOW A LIST OF FACTS.
If your goal is to let everyone know the exact GDP of countries, use a table not a graphic.

It is not a dilemma that people would see only a small difference in the colours of African countries because there is only a small difference in the GDP of African countries.

One last puzzle. Can you spot what’s wrong with this graph?

The problem is that again they’ve grouped things together when not necessary. But more importantly, look at the way the “Barely Dem” and “Barely GOP” are coloured. They are different to the others in that their centers are not coloured. This destroys the semantic meaning of the colours of the graph. It was originally that the colour represents the strength of the likelihood but not the colour AND how much white is in the center also represents it, adding complexity without adding any information.

One last error in this graph is that they have changed the size of Alaska. This is only confusing. The graph is being totally warped by the designer who felt that the shape of Alaska was awkward to work with.This makes the graph now entirely subjective. The designer could’ve also changed the shape of all other states – the integrity of the graphic is entirely destroyed by this decision.

Consciousness is not at the Core of Problem Solving

Learning to walk takes some time. There’s a reason for this: walking actually requires you to solve simultaneous equations. Yes, that’s right. It’s incredibly difficult math. It’s only recently that engineers have been able to build a robot that can walk on uneven terrain with four feet. I am yet to see one that can do so on TWO feet. There is Asimo but that only works on flat surfaces and so isn’t very impressive. To program a robot that can walk you need many gyroscopes in. Ray Kurzweil touches on this in Age Of Spiritual Machines
“To predict where the ball will go, and where the fielder should also go, would appear to require the solution of a rather overwhelming set of complex simultaneous equations. These equations need to be constantly recomputed as new visual data streams in. How does a ten-year-old Little Leaguer accomplish this, with no computer, no calculator, no pen and paper, having taken no calculus classes, and having only a few seconds of time? The answer is, she doesn’t. She uses her neural nets’ pattern-recognition abilities, which provide the foundation for much of skill formation.”
This assertion that she is not solving the equations is wrong. She is solving the equation but only in a way that doesn’t lend itself to being expressed in math. The solution to the simultaneous equations are in her brain unable to be expressed by her. Some people struggle with expressing emotions. I argue that we also struggle expressing the solutions of complex equations we solve.

To think it is pattern recognition is absurd. No two balls ever take the same path, how can it use pattern recognition? To transform the ball’s path into a previous pattern we’ve seen would be more computationally expensive than solving the simultaneous equation. It wouldn’t allow us to predict the path of a ball in windy conditions or when the ball has spin or anything of the sort.

The conscious mind does not give us unrestricted access to the computations our neurons are doing. Why would anyone believe it could? You can’t access the part of your brain that’s telling your heart to beat or your stomach to pump food. We can only observe a tiny fraction of our minds and how it functions. This makes many people uncomfortable because they identify with “their” conscious mind and enjoy living in the fantasy world in which they have absolute conscious control over their decisions, fueled by the illusion that is free will. If I say “why did you choose to play football instead of doing your homework?” saying “I don’t know” leaves many unsatisfied. As though not knowing is a reason to not have done that. There is a lack of consistency because if people were asked how they were walking (and express it in computer language or math) they would be unable to yet one wouldn’t expect that person to then stop walking.

I don’t fear doing something without a reason for it becoming consciously apparent to me and neither should you. You’re really blocking your own intuition and your greatest weapon – the unconscious (subconscious) mind. A recent article I read on this subject was about crows solving problems without planning their actions. I’m going to interpret the results differently. When we do things on the fly is called improvisation.  I think that what we call planning is actually just improvisation in a virtual world that we have constructed by imagining it. In other words, the same processes in our unconscious mind (the processes of solving the problem at hand) when we are in the real situation than the imagined one. It’s just that we are storing the improvisations we made in the virtual world in our memory so that once we encounter the real scenario we have already had a head start spending time improvising the answers. It is not that when we plan things it is our conscious mind overseeing this whole thing.

Now one mind say that my point is wrong because the conscious mind is the one that solves improvisation problems but this is not true. I see it as the unconscious mind solving problems then that solution rising up (“falling down “is perhaps the language I should have used) to the level of consciousness. The conscious mind then swiftly takes credit for solving this problem. It reminds me somewhat of the quote “good artists copy, great artists steal”. This quote was brought to my mind for unknown reasons but I still feel its relevant. I don’t need to be able to express justification for everything I do in order to be satisfied with it.

I’m guilty of separating the brain into conscious and unconscious parts. This isn’t quite right. Our brain is just one big mush of neurons and somehow it works.  To split it up in this way isn’t going to be very successful.

Our Displays Are 2D Not 1D: GUIs Need to Exploit This

Many applications do not take advantage of the fact that our displays can be arbitrarily wide. Almost all of them only extend their GUI in the vertical direction.

Of course, GUIs are shown in 2D. In 1D they would have a width of 0 and hence be invisible to the human eye. But the core of their design doesn’t take advantage of the horizontal dimension properly. Here are some examples.

Here Google uses only 1 column regardless of how wide your monitor is. It’s insane. The number of columns should increase just as the number of rows increases as the height of your monitor increases. Bing has the correct solution.

As you increase the width of your browser the number of videos per row increases unlike Google. This concept can be applied to other programs such as chat ones. Here’s what chat current looks like

Here’s how it could look.

There are two columns of text instead of one. The scroll bar would control both the left and the right windows. The left window is just an extension of the one on the right. If you were to scroll upwards 1 line of text, m would be at the top of the right panel and l would be at the bottom of the left panel.

The advantage of this is that the amount of information visible to the user is proportional to the area of the screen available rather than just proportional to the height of the screen. This is a fundamental improvement over the previous. One downside to having more columns of text is that the user may confuse the two but this is easily remedied by having a large space between columns and a border separating them.

This design could be applied to any list of text like Google’s search results. Though for that there is a better solution. Take a look at the website of Digg.

Digg does have multiple column but the numbers of columns is not dynamic. There are always three columns regardless of how wide your monitor is. They need to learn from Bing. This layout with a dynamic number of columns would be ideal for a site like Google. The top search result would be at the top left, the 2nd best would be on the next column in row 1. It would not be below it. This is because we read from left to right, top to bottom. Not top to bottom, left to right. It’s key that the items would be ordered correctly.

Accuracy is Everything

Accuracy is everything

If you want to become a good debator and an effective communicator, accuracy in your statements is essential. I make a conscious effort to avoid using words such as “really” or “actually” when it adds no meaning to the sentence. Instead of saying “1 + 1 is actually 2” I just say “1 + 1 is 2” because by using “actually” I undermine my sentences without it in, because “actually” acts an an intensifier in boolean statements (where as “very” is used in continuous statements “1 + 1 is very 2” wouldn’t make sense but “the sky is very bright” does make sense because the potential brightnesses take on a continous space) but this makes no sense. A statement is either true or false. It can’t be, no matter how much you want it to be, half true or mostly true.

I am not a pedant, but I am often called it. Another phrase I avoid like the plague is “too”. This word is misused to often. It should only be used for shortening sentences when the long form of the sentence is easily deduced. So the sentence “There is an amount of water being poured into that bucket such that water is flowing out of it and onto the floor” is simplified to “There is too much water being poured into the bucket”. But people rarely think about the long form of the sentence, they use the “too” shorthand so often that they don’t even bother to think about the longer form of the sentence they’re shortening. As a result, in their mind the long form of the sentence evolves to something like “there is an amount of water being poured into that bucket such that something bad is happening” and use too always in this sense, so it becomes meaningless when used in some cases. One example is “Don’t eat too much pie” which is short for “Don’t eat an amount of pie such that something bad will happen”. It’s a barren tautology almost. No one would ever do something that they thought was bad. By saying it, you do nothing to help the person – no information is contained in the sentence.

English is not rigorous but I take pride in meaning everything I say. I don’t use metaphors. I hate them. It’s just butchering my language. Stop it. Don’t make demonstrably false statements and then say “but it’s not false, it’s a metaphor”. Just because a sentence can be constructed without breaking the laws of grammar does not make it coherent. Metaphors are basically art but made it language rather than a paintbrush. I don’t hate analogies that help people understand things more by means of pattern recognition. I do find metaphors that are blatantly false annoying, such as “A lifetime is a day, death is sleep; a lifetime is a year, death is winter”. All of those are false statements.

Here’s an analogy that explains my position. It’s like I’m an air traffic controller and the local radio station decides that it will broadcast songs on the same frequncy I’m on. Sure it sounds great. But people will die as a result. I’m trying to communicate with pilots and you’re making this much more difficult by blasting your art over our communication. It’s sort of muddying the language I’m using. Take your art to another medium – it isn’t suitable to use English in this way. I don’t buy into the hand waving “it’s a metaphor” when I point out that the statement “Death is winter” is false… and neither should you.

“Literally” was a word that used to mean something. Sadly, so many people have begun to attempt to use it an an intensifier that it has started to lost all meaning. “She was literally the hottest girl I’ve ever seen” strictly means that the girl I saw had the highest temperature of all girls I’ve ever seen. It does NOT mean “She was the most attractive girl I’ve ever seen and I’m not lying”. This is basically the path English is going down by having so many people use metaphors. But at least people know when they’re using metaphors. So it doesn’t matter too much. Where as people who use ‘literally’ incorrectly are unaware of their errors.

The Nature of Computation

If you were building an AI for a game of chess it would be strange if you did anything other than modelling the game. But take a look at the universe we live in and the best equations to explain the reality. Some physicists will tell you that time does not exist. That time and space are really made of the same thing. It is my conception that if we put all the greatest minds together working on a chess AI, they would come to the conclusion that there is something more fundamental that the pieces and the squares. There is some more fundamental substance that they are both constructed from that would serve us better when creating an AI. Just as space-time  is named, we’ll call this fundamental matter square-piece.

This square-piece would allow a much more effective AI. An argument against this might be that we don’t know what reality is defined as, but chess *is* defined as pieces on a board obeying various rules therefore it can’t be anything else. This is wrong however. Just because a problem can be defined in a certain way does not mean it cannot be reduced to a simpler problem. Reducing the game so that the AI can make computations on the fundamental matter would be more effective.

I enjoy making wild statements that are difficult to verify (this seems to ruffle some feathers at times but I don’t care), so here’s another: our brains are actually operating on this fundamental square-piece when playing chess. This is how we can still compete current day chess AI that just iterates through potential scenarios on a weak processor that has a ‘mere’ 100 million transistors on it.

I think to exploit this square-piece a cellular automaton is needed. The space of algorithms that can be designed by humans is infinite in size but that does not mean it explores the entirety of the space of algorithms. There are many algorithms our brains could never comprehend that are essential to building intelligent systems. Instead of designing a CA, we need to search for one that does what we need. Play chess well.

The Weath Gap

The rich keep getting richer. This itself isn’t a problem. It’s just that if the wealth was distributed more evenly society would be a better place. There is one simple way to achieve this. By paying people according to how much money the company made in the past x days (where x is the number of days between payments).

The algorithm used would take a lot of designing but once made it could be used for all companies. It would be appropriate for any companies.

Reality is fluid. Many of our systems are rigid and therefore sub-optimal.It’s odd that people don’t really understand this. Suppose I say “worker A should be paid $19,435.24 this year” most people would react “that’s an oddly specific number, why not $19,000.00?”. Yet if I said “This large rock has a mass of 19,435.24kg” it would be met without the raising of eyebrows. It would be strange indeed if the mass of every rock I found was divisible by $1000.00.

It’s as though people are saying that I need justification for having a number which strays away from numbers with an accuracy of more than 2 or so points. This accuracy is their own system and doesn’t reflect reality. People basically believe that if you have a number with few trailing zeros like 19,435.24 then you need justification from straying from the solid, proven value of of 19,000.00. But that’s nonsense. 19000 in base 2 is 100101000111000. There is nothing special about numbers that have lots of trailing zeros in base 10. The number of trailing zeros a number has in base 10 has no impact on how much a worker should be paid. They’re not linked in meaningful way so to attack someone for suggesting a number with no trailing 0’s in base 10 is meaningless. There would be tempted to say “buy you cannot notice a difference between $19,000.00 and $19,435.24 and I would say “I can notice the difference between any numbers that are not equal because the just-noticeable difference does not exist“.

It is known that it makes sense to pay people more if the company is more successful since they contributed to it and there is more money available to hire better people. There should be an algorithm that automatically calculates the ideal amount to pay people. I don’t know exactly how that algorithm would look, but I know some of its properties.

First, it wouldn’t take the form of x + …. there would be no addition whatsoever in the calculation. It would all be based on multiplication and exponentiation. There would be no base rate. Reality is fluid, no part of it is rigid, everything is vibrating, and our calculations must reflect that.

Secondly, as input, it would only require the important of that persons job and the amount of profit the company made that year. As the importance of the job increases their salary increases and the dependency of their salary on the profits of the company also increases. So those at the top would lose more of their salary (as a proportion) than those lower down if the company showed a loss that year.

With the automation of many low skilled jobs, it’s inevitable that the rich will get richer. The skill level of jobs that can be automated will steadily increase as AI improves. First it was the people on assembly line. Next it will be stock brokers. Then it will be the programmers who made those stock buying algorithms. This will be the end of capitalism. People will revolt. When the top 0.1% controls 99.9% of the wealth a revolt will trigger.

I don’t know when or how this will happen, but it will happen. It is just the nature of capitalism. If you take any normal looking system and extend it out for a long time, its ultimate fate will always appear strange. Some say the fate of this universe is just dead. Entropy will increase so much that it will become impossible to support life. Other scientists say that the universe goes in one big cycle. After the big bang is the big crunch.

We Need A Revolution In Battery Technology

The desire for better batteries has never been greater. With so many mobile devices comes an even greater need to have reliable batteries with a much greater capacity. The best batteries we have are atrocious compared to gasoline and other forms of energy. Here is a graph from Wikipedia

The bottom left corner shows Lithium Ion Batteries. Their energy density is 0.72MJ/kg 1and 1.5MJ/l. Compare that to 47.2MJ/kg and 34MJ/l for gasoline.

Suppose we had a massive breakthrough right now that gave us a battery technology with those energy densitiies. Many impossible things would be very possible. It wouldn’t just be that our phones and laptops lasted longer, entire industries would change. Electric cars would immediately wipe out all the competition. That alone id a dignificant change.

Electric cars which last a long time are a dream. Much queiter for one, and more environmentally friendly. They’re not perfect- that electricity they’re using has to come from somewhere and in some cases it comes from a coal power plant. It is more efficient than allowing each car their own engine though, so it’s a big positive. Cities in China would be most impacted. Over there their cities are riddled with smog and electric cars would go a long way to solve that proble.

All those gasoline statinos would go out of business and that’s a good thing. They are inherently unsafe. To get more petrol is a dangerous job. A truck full of the stuff has to drive there. That’s crazy. There have been many incidents where a tanker has overturned and killed people in the explosion. There have even been incidents in poor countries like africa where people have gathered around an overturned tanker and tried to gather up the petrolium gasoline that has been spilled only to be killed by a huge explosion trigged by only a tiny spark.

Climbing mount everest would be much less challenging. The mose difficult aspect of it is the cost. If that was removed it would be tough. With batteries with high densities, clothes with heating built into them would work. Think how an electric blanket works except instead of being the shape of a blanket it would be clothes shaped and run on a battery. It would be fantastic. Of course it would still be very tough to climb mount Everest and would take a lot of training but that’s a lot of the difficulty taken out of it, dealing with the cold.

Tools that typically use gasoline or diesel would now use electrical power via batteries. Things like lawn mowers and leaf blowers. It would make them slightly quieter as there is no engine inside them but more importantly they would be more safe and more reliable. Dealing with oil is a dirty job.

I think that making these devices entirely electronic makes it easier for software to be used with it. Instead of having many buttons that do different things, there would just be a touchscreen on the device that had all the options you needed after you navigated through the easily understood menu. This means there’s fewer moving parts and hence increases the reliability of the device. In cars in particular there’s far too many buttons and knobs. All of that could go with the introduction of a touchscreen panel.

Rocket ships would still have to be powered by fuel. A battery could never generate the power needed for liftoff without being of a too large size. It basically needs a controlled explosion in order to get sufficient thrust for escape velocity.

It would revolutionize our world. The breakthrough would be undoubtedly one of the biggest breakthroughs of all time. We need a genius to help us solve our issues. Many scientists are working on this problem as we speak but none of them are bright enough to solve it. There have been small success stories such as making batteries with a greater power output but less capacity. More money needs to be put towards paying scientists to find materials and methods to reach this goal.

Working Towards A Prisoner’s Dilemma Solution

For those without knowledge, prisoner’s dilemma is basically this: there are two men in prison and they both worked together in a bank robbery but the police don’t have enough evidence to convict them. They are trying to get each of the prisoners to state that the other robber was involved in the robbery and give details. If neither of the criminals say a word, they each serve one month. If one criminal states the other one robbed the bank, that crinimal goes free and the other serves 1 year. If they both tell on eachother, they both get 3 months. The criminals can’t communicate with eachother, they are interviewed separately. What should the prisoners do?

 This is an unsolved problem in game theory. It’s very interesting. This isn’t however a 2 player game. It’s a 3 player game. The two crimininals are the police are all separate entities in this game. That’s akey distinction. In two player games theree is no dilemma, simply do whatever makes the biggest difference between you and your opponent.
 
 Here’s an interesting though experiment I came up with which may help with solving the problem. Suppose there are two people who are exact copies of eachother down to the atom. They are both placed in isolation in rooms with two buttons. One red and one green. If both press the green button both will recieve $1000. If one of them presses the red button and one presses green, the player who pressed red gets $5000. If both press red both get $0. What action should be done?
 
 The most important concept in play is that whatever button one person presses, the other must press two because they are exact duplicates of eachother. If one person has a given though process the other must have it too. Keeping this in mind, both players know that if they press red, they will both press red and if they pick green they will both press green. There is no scenario where one presses red and one green. This makes the solution rather simple: press green.
 
 Even though each player knows that the other player is pressing red, this is not sufficient justification for switching to pressing red in an attempt to exploit the other person because the other player will jump to the same conclusion.
 
 Now what would happen if the subjects were exact copies of eachother except for one atom? They both knew this. What should their actions be? They should both press green again. But there is a non zero probability that one presses red and the other green becauuse that 1 atom difference could change how a person reasons about the buttons and how he ultimately makes that choice. It’s incredibly unlikely.
 
 What about 2 atoms? And 3? As the differences in each player increases, the reaoning we used that the other person must do the same as us becomes weaker and weaker. Is it therefore more likely that both players press red as the differences between them increase. Essentially there is little guarantee that the other player will not betray them in a grab for the most money.
 
 One might think “Ah, the opponent thinks I’m very similar to him but I’m not so he will press green and I will press red”. This strategy would work. If one player thinks he is similar to the other, but the other does not think he is similar to the other, the player who thiks he is dissimilar would come out victorious on some occasions.
 
 If you know you are going to be in a situation like this, before going into the scenario instead of saying “press green” (that will be of little use) try to convince the other person that you aree very similar to him and think like him. This will allow you to press red when he presses green.
 
 This may go towards explaning why similar people trust each other more than disimilar people. There are many every day scenarios which are analogous to prisoners dilemma.
 
This sort of reasoning applies when there’s only just 2 players. Supose you live in a flat with a housemate who is similar to you. When a person leaves the flat they will leave their wallet with lots of money in at the flat. It would be easy for the other person to steal the wallet and get away with the crime. So why isn’t it done? Because it implies that the other person would have sloten their wallet. I must remind you that free will does not exist. The house mates are bound to act similarly to each other as a result of being similar. Thinking of the situation in a vaccuum is impossible. You can’t just observe the situation until one housemate leaves, then say “a rational actor would steal the wallet”. It is essentially that causality goes backwards in time. By stealing the wallet, you make it more likely that in the past your wallet was also stolen without your knowledge. Even if you have perfect knowledge of the past and know your wallet was not stolen, you can’t think of it like this because the reason it was not stolen was because it was determined that in the future you would not steal his wallet because he didn’t steal yours. You can’t go against this calculation of the future that was done if it was accurate. It’s binding. Free will doesn’t exist – you can’t use it to escape predictions of the future.

It Adds Up: Minimizing Delays In Popular Systems

Google gets somewhere in the region of 2 billion hits per day. It is essential that the Google page loads quickly. How much would it cost if the Google web page took an additional 0.1 seconds to load? 0.1*2,000,000,000=200,000,000 seconds would be wasted waiting for it to load. If we assume the average worker is paid $10/hour, that’s 200,000,000/60/60*10 = $555,555.56 wasted per day as a result of that 0.1 second loading time. So per year it would be costing the economy $147,222,222.22.

Most people overlook small differences in loading time, but it’s key. I’m disappointed to see that most OS manufacturers don’t optimize boot up speed. OS makers seem to care more about how many new features the new OS is going to have rather than shaving seconds off boot time and loading times which is far more important. Microsoft has spent massive amounts of resources on the Metro interface. It will take massive amounts of time per user to learn how to use it effectively. Not only that but each time the OS boots loading that huge interface will take more time. It’s a terrible decision.

UI designers are often unaware of how much time their poor decisions could potentially waste. Adding a single dialog box “Are you sure you want to do this? Yes. No.” when applied to millions of users has a huge cost as a result of wasting the users time. One might be tempted to say “But no ones going to notice that 0.5 seconds of their time was wasted” I would have to direct you to my blog post explaining that the just-noticeable difference does not exist.

It’s  time for UI design to become more quantitative instead of qualitative. Lots of tests should be done and the test subjects should be asked to do various tasks and this will be timed. If it takes longer on average then the new UI should not be added. I doubt Microsoft has even attempted to even do this kind of analysis. There’s no way they would add the abomination known as metro had they known just how bad it is. I haven’t used it myself but I trust that all the reviews I have read are correct to say that it is terrible.

Chess Is Dead

Chess AIs are now far better the top GMs (Grandmasters). This has dire consequences for the realm of chess players. It allows cheaters to use AIs to help them play. This devastates the game as it. Previously, cheating would have been impossible at the top level since it would be impractical for a GM to assist you playing. It is now possible to run a program on your smartphone that plays better than the best player in the world. If a person goes to the toilet in a chess match, how do you know that they aren’t putting the board onto their phone for analysis?

Chess online is also ruined. It’s impossible to know if you’re playing against a human or an AI. AI isn’t a plague but it’s more thrilling as a game knowing that you’re up against a human. I am in favour of AI.

It’s time for the GMs to find a new game to play. We have been exceeded by computers. There is no need to keep playing a game that computers have become better than us at. It’s a waste of your time. I don’t mind if you genuinely enjoy chess and continue playing. But as a form of art chess is no more. It is now solved by the brute force of computers.

I am learning to play Go because there is no AI in sight that could come anywhere near the top players on an 19×19 board. It’s exciting to know that the realizations you have cannot be beaten by a computer enumerating all possible moves.