It’s quite possible that in your early school days, you had to draw a line of best fit through some data by eye, with a pencil and ruler. Not necessarily a bad way to do it, but it’s not going to be the optimal line every time (unless you’re extremely lucky). Of course, these days, you can just ask a computer program to do it for you – no doubt many of you have used the ‘add trendline’ feature in MS Excel.

Depending on how advanced your mathematics/statistics education is, you may have drawn a ‘line of best fit’ (or regression line) using a formula. This formula gives you an easy way of working out the exact slope and intercept of your regression line, given information about the data you’ve collected. But have you ever wondered why this works? Where does this magical formula come from?

Some Terminology

I cooked up a graph to help you understand some of the things that I’ll be talking about. Firstly, we call y the regressand (or dependent variable) and x the regressor (or independent variable). This means that given some value of x, we want to see how the value of y depends on it.

If it helps, you can give x and y names. For example, x could represent years of experience at knitting socks, and y could represent the overall quality of sock produced by a particular granny. Each point on the graph represents the experience and output quality information of a specific granny.

Here’s where it gets interesting. There must be some underlying true relationship between x and y (it could be that there is no relationship, but that still counts). We don’t know what that true relationship is, because we only have a sample of n data points. Or, we don’t know the real relationship between years of granny’s knitting and quality of sock, because we can’t track down every single granny in the world that knits. That true relationship could be linear, or it could be something far more complicated. However, suppose I have magical powers and actually know the true relationship in this instance. I’ve represented that with a blue line.

Of course, not every point lies on that line. This is because there are probably other factors which we can’t observe that cause a ‘disturbance’ to the data. Your granny’s knitting ability may not only depend on years spent knitting, she may have had joint replacements that affect her motor skills or she may have inherited an instinctive genetic ability to perform a cross-stitch. Either way, the difference between the actual data observations and the true relationship is known as the disturbance (or error) term.

Let’s denote the disturbances of the true line as ε, the intercept (where it crosses the vertical axis) α and the slope of the line β. Then, for each observation i=1,2,…,n (or each granny), the true relationship between x and y is given by:

The equation of the true line is given by the first two terms on the right hand side. The disturbances are what takes us off the line and to a particular data point.

Now, as I said before, in reality, we wouldn’t know what the true relationship was. We wouldn’t know the ‘true line’, so we have to estimate it. This is what we call the ‘line of best fit’ or regression line.

In the graph, the fitted line is in red. As you can see, the line is a bit different to the true line, but this is because we are working from a sample and don’t have all the data. The fitted line allows us to predict, on average, what we can expect y to be given an x value. In terms of grannies, our line tells us that for a granny with p years of knitting experience, we can predict that she will produce socks of quality q. The equation for our fitted line is:

The ‘hat’ symbol on the letters means that they are estimates of the true values. Of course, if we wanted to get the actual data points, we’d have to move off the line. The distance that we move off the line is known as a residual. The residual is basically an estimate of the disturbance term, and so it’s not quite the same as the disturbance itself. So, to get the original value of y, we must add on our residuals to the fitted line for each observation i:

How can we draw a line that fits the data best? There are many ways of doing it, but the general idea is always the same. We want to try to draw the line so that we minimise the distance between our line and all the data points that we have. This, by definition, will be the line that best fits the data. The method we will use is the most common one, the method of least squares, invented by Gauss in the 18th century.

The idea is this: to get the line as close to all the data points as possible, we want to minimise the residuals. However, if we just add up the residuals, they’ll cancel each other out, because some will be positive (when the data point is above the fitted line) and others will be negative (when the point is below the fitted line). So we instead take the squares of the residuals and minimise the sum of them all. Nice neat idea.

In summary then, what we need are estimates for α and β such that the residual sum of squares (RSS) is minimised. All of a sudden, our problem has become a 2-variable optimisation problem, and so we can use calculus to solve it.

Deriving the Ordinary Least Squares (OLS) Estimators

The thing we want to minimise is the RSS, which is:

Here, I’ve just substituted for ε by rearranging the previous fitted equation. We can continue simplifying by explicitly squaring what’s inside the brackets:

Note that to simplify, I’ve used some shorthand to represent the sample means of x and y:

To minimise the expression for RSS that we’ve obtained, we partially differentiate with respect to the variables we want to get optimal values for:

The first simplifies nicely to give us the expression for the OLS estimator of α:

The second requires a little more fiddling:

Substituting for α:

Rearranging:

Or equivalently (and a bit more neatly):

And so, we’re done! We now have the OLS estimators for the case where there is one regressor. This analysis can be extended to allow for any number of regressors with the use of vectors and matrices, but I won’t go into this here since I think we’ve done enough for one day. Happy line drawing!

The Greeks were obsessed with the beauty of numbers. They liked everything to work out nice and neat, and this meant that their proofs and theorems were very elegant on the outside (though they incorporated a lot of imaginative and abstract thinking behind the scenes).

Unfortunately, this meant that they got a little irritated when things didn’t work out so nicely. One such instance came down to Pythagoras when, after championing his theorem about the sides of a right-angled triangle, he came across a problem in trying to work out the length of the hypotenuse when the other two sides were of length 1. That problem was √2. ‘Haha’, I thought, when I first heard that story many years ago. ‘Stupid Pythagoras’. Yet, the puzzle was not an easy one to solve.

Of course, you probably know that √2 is an irrational number, because this would have been something that you were told. This means that you cannot express it as a nice fraction, and, as a result, writing it out as a decimal number is also futile because it will go on forever.

However, to actually show that it is impossible to write out √2 explicitly is easier said than done. When I thought of it as being a nice proof to look at, it turned out to be a lot more difficult to crack than I expected! The proof emphasises the beauty of number theory, and will hopefully also make the point that there’s a lot more going on behind the things we take for granted than it may seem.

The Hard Bit

There is a preliminary proof that I need to do in order to prove that √2 is irrational. I’ll try to take this slow, because conceptually, it is not so easy.

I need to show that:

For any positive integers p and q, we have that p² ≠ 2q²

Eh? ‘Why should this be the case?’ you say. Good question. It’s not obvious, which is why we want to prove it. Try putting in some numbers for p. 3² = 9 but  9/2 = 4.5 which we cannot obtain by squaring a number. 4² = 16, 16/2 = 8 and 8 is not a square number either. Try this with more numbers if you want. Eventually, you might resign yourself to the fact that it probably cannot be done.

To prove this, we are going to assume that it is true first of all. In other words:

Suppose p² = 2q² for some positive integers p and q

If this is true, then it means that p² is even (because it is a multiple of 2). But if p² is even, then it means that p must also be even. It is quite easy to prove that squaring an odd number will always give you an odd number, so I won’t do this here (although if anyone is having trouble with this then leave a comment).

Let’s call the smallest value of p such that the above relationship is satisfied P. So, for this P, we have that:

P² = 2Q²

for some positive integer Q.

P is even, as we know that all values of p are even. Therefore, since even numbers are just multiples of 2, we can write:

P = 2M

for some positive integer M.

Now we can substitute in for P in the original expression, so that:

(2M)² = 2Q²

4M² = 2Q²

2M² = Q²

Can you see what this tells us about Q? Well, Q² is a multiple of 2, so Q² must be even. And from what I said before, if Q² is even, then Q must also be even. So, just like we did with P, we can write Q in terms of some positive integer N:

Q = 2N

Now, we can substitute that into the previous expression involving M and Q:

2M² = (2N)²

2M² = 4N²

M² = 2N²

Want to guess what’s coming next? M² is a multiple of 2, so M² is even, and therefore M is also even! Does this expression look familiar? The relationship between M and N is exactly the same as the relationship between P and Q.

This means that M is another possible value for p. But what is M? Let’s look back up our equations. We defined M such that P = 2M. Hence:

P > M

Uh oh! We have a contradiction! We defined P as the smallest possible value of p. But M is also a possible value of p, and M is smaller than P. So P cannot be the smallest possible value of p!

If we use this argument recursively, it will go on forever. This means that there is no smallest value of p because we can always find one smaller. But if you have a set of numbers, there has to be a smallest one… unless the set is empty! Therefore, we have just shown that the set of numbers that satisfies p² = 2q² is empty. In other words, this relationship is impossible to satisfy!

The Easy Bit

That was probably a bit difficult to get your head around (if not, then you should take up mathematics at a high level if you aren’t doing so already!). Now, we can prove that √2 is irrational, and this is actually quite easy in comparison.

A rational number is one we can write as a fraction p/q for some integers p and q. Of course, since √2 is positive, p and q must also be positive. So what we want to prove is that:

There exist no positive integers p and q such that √2 = p/q

As we did with the previous proof, the way to approach this is by contradiction.

Suppose there exist some positive integers p and q such that √2 = p/q

Squaring both sides gives us:

2 = (p/q)²

2 = p²/q²

2q² = p²

Aha! We have proved earlier that there cannot exist positive integers p and q such that this equality is satisfied. Therefore we have a contradiction. Hence, our assertion that √2 = p/q must be false. Of course, this means that √2 is not a rational number. Thus, √2 is an irrational number!

And with that, we’ve opened up a whole new realm of numbers – ones that we need to use but that we can’t write out explicitly. Isn’t maths exciting?

The Problem with iWeb

iWeb was actually one of the reasons that I was able to start this website with very limited prior knowledge of web design. I liked it. It’s very unintimidating for the newcomer, and that’s clearly what Apple were going for. I’m much happier and more comfortable when I can see how everything is going to look and how everything connects visually. I also like making little tweaks on the fly. For this iWeb was much more accessible to me than, say, something like Dreamweaver. It’s effectively a website-oriented version of PowerPoint, or a desktop publishing program.

Pretty on the outside. Horrible under the hood.

However, quite early on, I spotted a few very strange behaviours when I tried to create content. iWeb likes to give you the freedom to do anything. “How wonderful! Why can’t I do that with practically any other website design application?” I thought to myself. Turns out there’s a reason why you can’t.

Firstly, I was using some very cool fonts for parts of my site. Indeed, the in-built theme I started out with used Futura for pretty much everything. Unfortunately, this is not a web-safe font (and most non-Mac users wouldn’t have the font on their system anyway). Luckily, I changed this quickly, but I was rather confused as to why there was not a restriction put in place regarding font selection. What was even worse is that I started putting drop shadows on titles, and that screws things up to the point where text is saved as an image file! My first site creation attempt was a slow and buggy mess – and that was before I tried viewing it on Internet Explorer 6!

Secondly, the impression I got that I could drag and drop anything anywhere was misleading. Once the site was published, I ended up with shapes being split up and scattered across the page, and images overlapping text instead of the text wrapping around it.

Both these cases are examples of the wool iWeb puts over your eyes with regard to how web pages actually function. Because web pages are designed in terms of style sheets and frames, manual placement of individual elements wherever you want for each individual page should not be possible without some horrible and inefficient coding. And horrible and inefficient it all was.

iWeb treats each page as an individual – not as part of a template that runs throughout your site. This means that to create new pages in a unified style, you had to keep duplicating pages. This wasn’t much effort in itself, but because iWeb treats each page as unique, even if it is following exactly the same style as all your other pages, it means that common elements are stored and loaded separately for each page.

As an example of how silly this is, consider my signature image in the header of this page. It appears in every page (and in the same place) so the smart thing would be to remember this positioning and have one copy of the image on the server. Then, that image could be referenced for each page load. Because it is the same image, your browser wouldn’t have to keep reloading it, as it would be stored in your local cache. Instead of this process (which is what happens now), iWeb would save a new copy of that header image for each and every page I had on my website! This means that every time you navigated to a new page in my site, it would have to pull out the same image but from a different location. As far as your browser knows, this is a completely separate image, so it has to fully load it each time! This happened for every single image element in my pages, meaning that my site was horribly slow to load and used up far more storage space than it should have.

Finally, iWeb was designed to tie in closely with MobileMe. It meant that Apple wanted you to pay an extortionate yearly subscription fee for hosting your site. In return for this, your site would function with all the bells and whistles that they intended. I had no intention of doing this, of course, so I had to sacrifice automatic FTP updating and things like an inbuilt commenting system – something that is practically essential in a blog-based site.

Alternatives to iWeb

I stuck with iWeb for a long time for a few reasons. Firstly, I’d reached a steady state where I’d learnt how to get what I wanted out of the program by invading it a little. I discovered the deeply buried theme files and edited them to my liking. I’d learnt what fonts to use and how to avoid putting things in my pages that would create major slow-downs. The limitations and inefficiencies were frustrating, but I somehow hoped that some of these issues would be addressed even slightly in future program updates. Once I heard that iWeb was pretty much dead, and that there was therefore no potential for improvement, I thought it would be a wise time to ditch it altogether. I knew the process would be painful initially, but I was looking forward to having a much cleaner and more functional website.

Hence, I started looking at some alternatives. I wanted something that would allow me to create and maintain a site with dynamic content easily, which meant something with a large visual editing component. However, with the knowledge I’d amassed over time regarding web design, I was less afraid of dabbling with a little code if necessary.

Desktop Based Applications

Naturally, the first applications I tried to look for were direct iWeb equivalents. I tried a few of them out to see how easy it would be to customise my site to my liking, and what sorts of new things I’d be able to do (as well as the obvious issue of how easy it would be to transfer over a whole other website).

RapidWeaver

This was the first application I found when searching for an ‘iWeb killer’ and it’s one that I’ve come across in the past. It is less of a WYSIWYG experience compared to iWeb, but one that allows your end product to be more standards compliant. However, I found it to be way too fiddly to get things looking right. Whilst there are a lot of customisation options, you are still bound to built-in themes. These can be edited in a similar way to iWeb’s themes – via the back end. However, this usually involves editing chunks of code, which can become tricky and unintuitive.

Failing this, there are a lot of ‘plugins’ that can be downloaded for RapidWeaver that allow you to add tables and frames to your page. However, I found it strange and silly that many of these things aren’t built into the application itself. As an example, without the ‘Stacks’ plugin, there is no easy way of arranging your page into columns or framed segments. This wouldn’t necessarily be a problem on its own, but everything that even marginally improves RapidWeaver’s functionality must be paid for. This includes themes. Just to recreate my iWeb site on RapidWeaver, I would probably have had to spend in excess of $100. iWeb, whilst unwieldy, at least did it all for free.

Yup, time to empty your wallet.

Sandvox

Put simply, Sandvox is ‘iWeb Plus’. It’s pretty much exactly like iWeb, but with a few more features. It also generates cleaner code. For someone familiar with iWeb, it was instantly usable. However, I came across niggles in designing pages that were somewhat similar to RapidWeaver. Still, I liked the application, but I never really felt like it was much of a step up from iWeb, and there wouldn’t have been much point in expending the effort of migrating my site to practically the same program. It would have been like moving from solitary confinement to solitary confinement in a room that was a couple of square metres larger. Sandvox doesn’t really give you very many possibilities of expansion beyond the iWeb experience, and is certainly less advanced than RapidWeaver.

Goldfish and Freeway Pro

These applications gave me a much more desktop publisher-like experience than the former two. This was nice, as it appeared to give me more flexibility. However, I found that the freedom was actually a bit too great. I felt like I was designing isolated sheets of A4 rather than a contiguous, flowing web page. These tools were probably not designed for dynamic content such as blogs, and so it seemed like I could only really create a blog by creating individual static pages and manually linking pages together. This was actually a step down from iWeb in terms of functionality (though the page generation may well have been cleaner)!

Remote Solutions

RapidWeaver seemed to be the best option as far as the desktop clients were concerned, but I still wasn’t prepared to put my faith in it entirely. As I had pretty much exhausted the list of programs that I was after, my next step was to look at content management systems that were remotely stored and edited, rather than locally on my computer. I never really liked the idea of online editing, but there’s no harm in giving these solutions a try.

Blogger, WordPress.com, Tumblr etc.

I’ll give a brief mention to these ‘ready-made’ blogging infrastructures. I considered them when I first created this website, but discounted them almost immediately because I didn’t like the lack of control I was afforded. I would be lumped in with millions of other users with no real means of differentiation or sense of identity. A blog, which to me is a form of expression, becomes redundant when you surrender the extent to which you can express yourself. This is the reason why I still look down upon these platforms, and why I wasn’t going to consider using them this time either.

WordPress

I’d mucked about with WordPress in the past, and although I liked it, this was at a time when I was less comfortable at fixing and playing around with code directly to get what I wanted. At the time, WordPress was on version 2, and quite a bit less intuitive than it is now. (Note that WordPress is a content management system that is found on WordPress.org, but WordPress.com is a closed version of WordPress that is designed to function as a fixed, ‘ready-made’ blogging tool like Blogger)

Once I tried it again, I knew it was the right direction for my site. It is completely flexible, as you are directly in touch with the source files right from the get go. At the very least, you need to understand how everything pieces together so that you know where to look if something goes wrong, or if you want to change anything. The great thing about this is that it forced me to get my hands dirty and play around with code. By doing this, you get a real sense of the things you should and shouldn’t do when designing your site.

Like RapidWeaver, WordPress relies on themes and plugins. However, because WordPress is open-source, there is a huge support base for all related things on the internet, and almost every add-on or bit of code is freely available. Once you get more proficient at PHP and CSS editing, the only real limit to the things you can do is your knowledge of the code and your imagination. This is the kind of open-ended potential I was after. If you visited the old site, hopefully you’ll appreciate the difference!

Drupal and Joomla seem to be the two main contenders to WordPress’ online content management system throne. I haven’t really looked into these two enough to be able to comment on them, but I do feel that these types of open-source, remote packages represent the best option for those looking to make an individual and powerful website that is very easy to maintain and update with rich content.

What next for Apple and website creation?

One word – iCloud. Apple announced this as the replacement to MobileMe a few months ago, and it was this that effectively signalled the end of the road for iWeb. As I’ve discussed earlier, iWeb really creates horrible websites from a usability standpoint, and this goes quite against Apple’s own ‘clean yet functional’ design philosophy. Although you’ll still be able to use iWeb to create your own sites as long as you have the application on your Mac, the fact that Apple have stopped including it in new Macs with the release of OS X Lion suggests that they really want you to stop using it, and won’t really offer any support to people who run into problems with it.

One option is for Apple to call it a day when it comes to web/blog designing tools. They may just accept that there are enough good alternatives out there for them not to get involved. But does that really sound like Apple to you? After all, if anybody is capable of consolidating available technologies into a neat and elegant package, it’s Apple.

With iCloud, I think Apple may well go the way of self-contained blogging hosts like Tumblr and Blogger. By ditching the application-based method of content creation, they can make sure that everything runs optimally and smoothly within their own environment. This includes things like proper indexing, search engine optimisation and so on. With no software on the user end, it makes it easier for Apple to keep themselves up-to-date, and also gives them the degree of control that they desire over how people should and shouldn’t go about doing things.

Apart from the fact that they are easily capable of creating something that looks and feels a lot more polished than something like Blogger, the other big draw to a platform created by Apple is down to the fact that so many people are already part of the iTunes infrastructure as a result of the iPod, iPhone and iPad. Because of this, and because of the syncing features that Apple are planning to offer with iCloud, they can allow you to access this repository of data to add dynamic content to your website. For example, people could have their top 10 most played songs on iTunes listed on their front page, and this list would be dynamic as it would pull data from your desktop and mobile versions of iTunes. It would allow you to put a map with a live geolocation flag indicating your position via your iPhone. It would allow you to post new content easily from any iDevice. It could pull up your current availability from iCal and the calendar on your iDevice.

Yes, all of this technology is currently available, but it will be much easier to implement successfully if everyone is already part of the shared information infrastructure – which is certainly the case with iTunes.

All I can say is – watch this space. If this is the direction that Apple choose to go in, then it could even give social networking sites like Twitter a run for their money. Whilst I’m happy with my decision to switch to WordPress, I’m also interested (as always) to see how the future of content creation will unfold.

Not only do they paint a very interesting picture, I think they do a good job of summarising the headaches (and also some excitement through curiosity) that economists have been experiencing in the last couple of years.

There were two graphs that caught my eye in particular. The first one, looks at debt in the UK from private institutions and households, and compares this to public debt.

There are 3 obvious facts we can get from this graph at first glance. Since the global financial crisis:

1. Companies and households in the UK have massively increased savings (and most of this seems to have come from households).
2. The government has massively increased debt, almost to an equal magnitude as the private savings increase.
3. The underlying trend of a small financial surplus seems to have remained stationary for the rest of the world.

The second graph is a little simpler. It shows that real income in the UK has dipped a little since the crisis, but the overall trend, whilst flattening slightly, doesn’t appear to have been affected hugely. The big message here though, is that consumption has plunged since 2008, and has remained low since 2009. It is now well below real income, where it was previously roughly in line with it.

Why is this significant?

Well, the other bit of information missing here is the Bank of England’s interest rate. Interest rates have been at an historic low of 0.5% since March 2009 (see graph below). According to standard consumer theory, a lower interest rate reduces the yield we get on our savings, whilst simultaneously decreasing the cost of borrowing. So, one would expect that following such a dramatic reduction in interest rates, consumption would have risen (or at least not fallen as much). Indeed, there was a small recovery during 2009, but that seems to have worn off.

Mike vs Mike

It’s pretty difficult to think about the album without taking a close look at Mike Mangini’s new drumming, simply because Mike Portnoy had shaped Dream Theater’s rhythms for decades. Mangini gained a bit of fame in the 90s and early 2000s with Steve Vai. Whilst he is the better technician of the two by miles, I’m not a big fan of his tone.

Tone is something you hear guitar players talking about all the time. Every player wants to find exactly the right tone for his guitar, and this can involve years of finding the right combination of equipment. In addition to this, a lot of tone comes from the individual. How you move your fingers has a dramatic impact on how notes come out of the fretboard. Although it is not talked about as commonly, drummers have exactly the same individual influence on their tone too.

Portnoy’s fat, chunky style combines speed and decent technique with very solid, groove-oriented playing. I take particular enjoyment from listening to Dream Theater’s drum parts. They stick in the mind as much as a guitar solo would.

However, I cannot say the same of Mangini’s parts in A Dramatic Turn of Events. Firstly, his sound is very ‘weak’. This could be the result of a poor recording arrangement (the production values on A Dramatic Turn of Events are pretty bad, similar to those found on Octavarium). But if you listen to Steve Vai’s Alive in an Ultra World album, the drum tone is very similar. This leads me to believe that it is just a consequence of his playing style.

Superior technique does not a better drummer make. There’s something instinctive about the notes you choose to play or the drums you choose to hit when you hear some music, regardless of what you are able to play. I felt that Mangini’s note choice just didn’t seem right to my ears when I’d heard him before. In the opening seconds of the album, whilst I waited for the drums to kick in, I was left feeling quite disappointed when they actually did. More evidence that Mangini’s sound is not to my taste came my way many years ago when I saw Virgil Donati playing with Steve Vai on Live at the Astoria and really stamping his authority by improving on Mangini’s old parts in every way.

Throughout the album, the feeling I got from the percussion was very robotic. Almost like the drummer was a ‘hired gun’, given a wage for his labour and then sent home. The drums just do their job in keeping the songs going, and although they are flawless in doing so, they don’t add any new dimension to the music. This can be qualified by word that a lot of the drum parts were set before Mangini came into the band and so he may not have had much room for experimentation. If this is true, then many of my comments on the drumming may not hold. But if it is, then the question would be: why wasn’t Mike allowed to experiment a little more?

On a slight tangent, I was quite disappointed whilst watching the documentary that followed the band on their search for a new drummer. Virgil Donati was trying to innovate and improvise a lot of part to traditional songs, and it seems like this approach was shunned in favour of staying in a zone that is more familiar to the rest of the guys.

My view on the drummer issue is that Dream Theater are lacking without Mike Portnoy, and Mike Portnoy needs Dream Theater to showcase his talents to the fullest.

The Songs

On The Backs Of Angels kicks the album off with a trademark heavy and technical instrumental section after a bit of a build up. There are a few things that strike me about the sound of the album already. Firstly, as mentioned in the previous section, the production values are awful. I’m not sure what’s going on here, but in comparison with the last couple of albums, it’s a huge step down. Octavarium had a similar muddiness going on with it too. Secondly, the choice of drum grooves doesn’t please my ears too much. Finally, the lyrics and vocals seem a little ‘forced’.

However, this song is one of the better ones on the album. The guitar pre-chorus at 3:40 is beautifully executed, and Jordan Rudess’ synth and piano wizardry is breathtaking. It’s the closest thing to a modernised version of Pull Me Under that I can think of.

In general, this song is a good representation of the direction that DT has gone towards in the past few years. Good technical sections let down by a very stereotypical and cliched metal chorus, as well as John Petrucci trying a little too hard with his solos, to the point where some musicality is sacrificed in favour of the ‘look how fast I am!’ effect.

Build Me Up, Break Me Down starts off with a riff that sounds a lot like something out of an Alter Bridge song. This song is another good example of the run-of-the-mill metal that seems to have seeped into Dream Theater. Listening to the verse with its processed vocals and then listening to any song from Images and Words makes me want to cry a little. Nothing in particular to like about this song, apart from possibly the anthem-like chorus which, although a little cheesy, is somewhat infectious after a few listens.

Ahh, a piano opening that is reminiscent of Muse’s Butterflies and Hurricanes kicks off Lost Not Forgotten. The initial drum groove reminds me a lot of Virgil Donati’s work. It’s well executed, but the kick drums sound a little too thin and poppy, lacking any real body. The early harmonies and unisons between keyboard and drums really showcase what I mean by losing some of the musicality over just showcasing technique. The sweep picking sections remind me of DragonForce – or in other words, a pinball machine crossed with a Super Nintendo. This is not a good thing in my book (although the Super Nintendo had its fair share of brilliantly arranged music). The later guitar and synth solos are also quite forgettable, although technically brilliant as usual.

Halfway through the song, there is a really nice bass break and an uplifting staccato guitar and vocal melody. Speaking of which, John Myung really doesn’t seem to get to flex his bass muscles as much as he should these days.

This Is The Life is a powerful rock ballad, in the same vein as something like The Answer Lies Within from Octavarium. An emotive song, it sets a great mood. No objections here – this is one of the things Dream Theater continues to do very well.

You don’t particularly want to start a song with something that sounds rather like a frog vomiting quite violently. Bridges In The Sky, however, manages to do just that. I’m not sure what they were thinking here. The choir section following it does do a good job of setting a dark mood for the song overall. The riff is quite reminiscent of the 7 string chunkathon that is Dark Eternal Night. The vocal lines in this song are particularly strained, though this is likely the fault of the low key rather than James LaBrie’s voice.

Other than this, it is really a case (once again) of rock chorus and pretentious instrumental sections. The continuum solos are getting a little repetitive, Jordan.

Another Alter Bridge riff? Nope, this time it’s Outcry. It gets better (read: worse) when the drum machine kicks in, and Alter Bridge turns into Linkin Park. Another cliched chorus. Sigh. But then we have the extended instrumental section. Whilst another frenzied encounter, there’s something about this one that actually sounds musical. It is abstract, yet well orchestrated. Systematic Chaos – to use a cheap pun. Aside from the opening, Outcry actually develops into a solid effort, the kind that people have come to expect from Dream Theater. Maybe we’ve been spoilt by the abundance of them in previous albums.

Far From Heaven is a piano and vocal ballad, following the style of Wait For Sleep and Vacant. However, the lyrics aren’t as chilling and the delivery not as sweet. It’s pleasant, but very forgettable.

There seems to be pattern emerging in the intros to the songs on this album. They all seem to remind me of other bands! Breaking All Illusions starts with a string chord progression that sounds a lot like Coldplay’s Viva La Vida (Somewhat ironic, since this song also attracted a lawsuit from Joe Satriani for copying the lead from If I Could Fly). The chorus uses this chord progression too, and actually, hardly any of the choruses on the album have lifted the song in the way they’re supposed to.

However, as with Outcry, this song actually develops into something brilliant. In fact, it’s the stand-out masterpiece of the album by a long, long way. This is cemented by some of the best lyrics on the album and very addictive riffs. Truly progressive, the song shifts gears multiple times. As an example around the 5:00 mark, you hear a delightful Middle-Ages style keyboard interlude, followed shortly by piano and organ breaks, as well as a disco-style guitar groove.

But by far the best thing in this song, and on the entire album, is John Petrucci’s guitar solo. He always seems to pull one out of the bag when you start losing faith. This is the Dream Theater of old. This is what we all like to hear.

In recent years, JP’s tone has become cleaner to the point of becoming slightly sterile. I can understand why to some extent: because he’s utilising more speed and technique these days, ‘clean distortion’ is what’s required to stop the notes bleeding into each other. However, I really miss the character of the ‘dirtier’ crunchy distorted tone that he used on older albums.

We end the album with Beneath The Surface, another melodic ballad. This one incorporates some very nice guitar work, and the vocals are up to par. The synth solo sounds a little out of place on this track, and the strings are a little over the top later on in the song. It is a beautiful way to end the album though.

Overall Thoughts

For me, Dream Theater peaked around the Six Degrees of Inner Turbulence album 10 years ago. Octavarium was their last true masterpiece in my opinion. Since then, a lot of the traditional progressive elements that defined Dream Theater have been washed away.

A Dramatic Turn of Events is a mixed bag. At times, it brings back some old-school progressive moments, but these moments are marred by completely forgettable sections that get lost in the sea of modern metal material. Dream Theater has certainly lost some of the production and ‘big picture’ orchestration that I think Mike Portnoy brought to the table. Without him, the technique remains, but it seems to be lacking direction.

But things were sounding a bit fatigued to me even in previous efforts when Mike was still around. Maybe, as Mike felt prior to leaving the band, the machine is a little tired of churning out albums at regular intervals. His suggestion of a 2 or 3 year break doesn’t seem like a bad idea. Even if the talent is still there, sometimes you just need time for reflection before you get hit by inspiration once again.

I’ll leave you with this tongue-in-cheek take on the album:

Physically, you are training your muscles to develop ‘muscle memory’. In other words, by repeating the action over and over again, you are essentially ‘programming’ it into your muscles so that it feels natural. If you have gotten this far with any bit of physical skill, you might realise that you don’t actually need to think as much about what you’re doing once you get quick enough.

This is what I’ve been thinking about, and I want to offer a few of my thoughts on the matter. Apologies in advance for the lack of coherence to follow…

What goes through our mind?

The point that interests me the most is the one I just mentioned. I find it a little strange that we actually think less about something the better we get at it. Surely, in the process of acquiring greater skill, wouldn’t you think that the brain is doing more work because it has to handle more information in a given space of time?

What seems to be going on here is a form of subliminal processing. It reminds me a lot of reading. Remember when you were a kid and learning to read? You might have followed each word with your finger and pronounced each word carefully by combining the individual letters in it. How about now? I’m sure you can just look at the word and subconsciously know what it is without actively having to look at each letter. In fact, you may have come across the famous Cambridge study which found that people only need the first and the last letters of a word to be in place to recognise the rest of it. For example:

The fcat taht you can udnretnasd tihs whiutot mcuh dfiflcituy is qtiue azmiang, dno’t you tnihk?

The same sort of thing seems to be going on when you’re internalising some sort of muscular motion by way of playing the guitar.

Fast Hearing

Returning back to music (and I’ll use the guitar as my example here), suppose you’ve just learnt a 4 bar riff inside out. You can play it at practically any tempo without any errors. If I asked you to play it now, would you sit there and think ‘hmm, I need to play an A and then move my finger 2 frets to play a B…’? You might have done that when you were first learning it, but you wouldn’t do that any more. Instead, your fingers would automatically get into position and your brain would do the processing in the background. You would have become so efficient at the process that the amount of conscious thinking that is required by you is minimal.
And in fact, I think that this is a very important step in being able to play something quickly and accurately. The brain just isn’t fast enough to actively think about each note separately. You will only become fast when you can relegate those thought processes until they become as natural as breathing. Think of The Matrix if you want an exaggerated version of the kind of thing I’m thinking of! I would conjecture that the faster the player, the lower the level of conscious activity in the brain will be. Efficiency of the mind, if you will.

The same goes for hearing music. When you play the notes, you will likely only ‘zone in’ at key points in the riff. You might consciously pay attention once each bar at places where your brain deems it necessary to focus in order to get your bearings right. It’s somewhat of an ongoing calibration process that only needs to come to the forefront when something significant (physiologically or musically) is about to happen in the piece. As a result, you won’t actually ‘hear’ each note, but you’ll intrinsically know whether it’s right or not.

Ever notice how the listeners of more complex genres of music are largely comprised of musicians in their own right? I think this is far from coincidence and links into this subconscious ‘fast hearing’ phenomenon. Where the untrained ear just hears a blur or flurry of random, seemingly unrelated notes, those who have this ability to process quickly in the back of their minds seem to be able to pick out the relevant parts and make sense of the aural information coming their way. I know that in my case, I’ve gone from thinking that a Red Hot Chili Peppers solo was unfathomably fast and incomprehensible to being able to mentally pick out runs that Dream Theater use in their music, just by internalising a lot of the extraneous information that comes with playing certain things on the guitar.

OK, so is there a point here?

Well, yes and no. I’ve pointed out something that I find quite fascinating: the way our mind becomes proficient at something by effectively minimising the level of conscious effort required. To me, it seems like a very fundamental optimisation exercise, and is a great example of how nature seeks to simplify and streamline everything as much as possible (which, as a side note, I think is why mathematics captures natural processes so effortlessly).

Although this information is not going to make you the next Michael Angelo Batio, I think that understanding the principles by which our minds work is important if we want to achieve a higher level of proficiency in anything we choose to do. At the very least, I hope this brain dump has provoked a bit of curiosity!

Once upon a time, the great country of Englonia was ruled by the benevolent leader Neelios. He was famed for his mathematical methods of problem solving, and peasants and landowners alike would come to Neelios for justice and fairness to be calculated for them. Of course, his success was, in part, due to the strong support he received from his five noble and wise aides.

Drove ‘the Car’ was in charge of healthcare and had almost supernatural healing powers for those especially in need. Nijller was a master craftsman. There wasn’t an arm, a leg or an antique table that he couldn’t slice up and restore to its original glory. Natalya was a master of communication and instilled a great love for language throughout the nation. Legend has it that she was once a cat. However, to this day, nobody knows the truth behind the mystery or where she actually came from. V-Ray was a cyborg. A keeper of the peace and master of justice, he enforced and optimised the law. Finally, Aware was a master of aesthetics and design. He also moonlighted as the court jester.

After many years of harmonious leadership, Neelios unfortunately became very ill. According to Drove’s diagnosis, there wasn’t much time left for him. Not wanting to leave things unsettled, Neelios had to decide how the 48 counties of Englonia would be allocated before he died. With his last gasp, he announced how the country would be split:

“I have decided what will become of Englonia after I depart. I give V-Ray control of Greater Londonium, our nation’s capital.

Of the remaining 47 counties: Drove shall receive half, Nijller will receive a third, Natalya shall get a twelfth and Aware will get a sixteenth.”

And with that, Neelios passed on. His corpse was traded overseas for chocolate.

—————

Everyone was very excited by the prospect of ruling their own lands. Whilst V-Ray toddled off to Londonium, the remaining four were left to think about the delegation process. They sat around a round table and contemplated over cappuccinos and chocolate croissants. After a short while, Nijller scrunched up his face. The others asked him what was wrong.

“Well” said Nijller, “I’ve just calculated our shares of the counties. It doesn’t look good.”

Drove agreed, but Natalya and Aware weren’t too sure why. Maths wasn’t their strongest suit. Drove explained the issue to them.

“Our shares of the 47 counties were: 1/2, 1/3, 1/12 and 1/16. However, when we work out how many counties that means for us each: I would end up with 23.5, Nijller would get 15.66666…, Natalya would get 3.91666666… and Aware would get 2.9375!”

Natalya sniggered because apparently Neelios wasn’t as great at calculations as he thought. But amusement quickly turned to bickering and frustration as the quartet argued about how this would work.

—————

Whilst Nijller was trying to cut Cornwallinium into thirds, Aware (who was quite unaware of the sheer magnitude of the task surrounding the others, and who had said nothing thus far) exclaimed,

“What if we ask V-Ray if we can borrow Londonium for a bit?”

“You’ve gotta be kidding me!” cried Drove. “You think we’re going to kick him out of his only county? The one Neelios specifically held for him? I don’t think that will go down too well…”

But Aware was on to something.

“We’d give it back to him tomorrow!” he replied.

Whilst Drove and Nijller proceeded to give Aware wedgies for his absurdness, Natalya began to think about his crazy scheme.

“Hang on!” said Natalya, with some authority. “If we borrow Londonium, then we’d have 48 counties to divide right? I think that could make things easier.”

She called V-Ray to explain the situation.

“Okay, I’ll let you borrow Londonium. But if I don’t get it back tomorrow and find out you’ve divided it among you, I’ll kill the lot of you.” V-Ray said, worryingly calmly.

He was a cyborg after all. Nobody wanted to mess with those bionic biceps. Natalya agreed.

After calming down Nijller and Drove (and sorting out Aware’s underwear), Natalya proceeded to explain why she had done what she did (it apparently wasn’t because she used to be a cat).

“We have 48 counties now. Drove, you get half so that’s 24 for you.”

“Great!” said Drove.

“Nijller, you get a third, so that’s 48/3 = 16.” Nijller could dig that.

“I get a twelfth so that’s 48/12 = 4 counties, and Aware gets 48/16 = 3 counties!”

Natalya was delighted that her plan was so effective, and blew raspberries at Drove and Nijller. The pleasure was short-lived, however, as the fear of death by V-Ray’s v-ray set in.

Nijller started doing some sums.

“24+16+4+3. That’s exactly 47 counties! Perfect! There’s exactly one left over!”

Natalya quickly called V-Ray to return Londonium to him. All that was left was for the four to decide who got what. At least they had a lot more chocolate to help them through that decision…

—————

And so, the problem had been solved. The famous five lived happily ever after and made their counties even better than they were before. Englonia had turned into a haven, and it was all thanks to some clever mathematics, a little bit of inspiration and a lot of chocolate.

The End

I’d noticed recently that a few people had mentioned the expensive cost of housing in relation to the average salary in their area. So I was interested to find out which places were particularly bad in the grand scheme of things.

To do this, I picked some major cities from around the (reasonably developed) world. I used payscale.com to get data for the median salaries for these cities, and predominantly numbeo.com for data on housing.

Methodology

Initially, I thought I’d find out the average house price for each city by looking at country-specific statistics websites. Unfortunately, what seems to be a fairly easy figure to work out is very difficult to come by, especially in the less developed countries within my sample! Many countries don’t seem to have great statistical reporting, and many others had independent firms with research on house prices that you had to pay vast amounts of money to see. For the data I did find on average house prices, it was often unclear how it was calculated, so it was difficult to compare figures credibly between countries.

Luckily, I managed to come across a source (Numbeo) which gave average house prices per square metre for individual cities. This way, I could compare equally sized properties between them. I’m still not too sure about the accuracy of the data sets, but this was the fairest comparison I could manage with the data I had.

Numbeo gives two prices per square metre: one for properties inside the city centre and one for properties on the outskirts. Not to choose one over the other, I averaged the two to come up with an overall average per square metre price for each city. A little research showed that a good ball-park figure for the average size of a property would be 100m2. Of course, the average property size will vary between cities. For example, congested Tokyo has very small apartments compared to the much larger places that you’d find on average in many US cities. Still, using a fixed area makes it easier to compare costs of living more accurately.

Once I had the average price of 100m2 worth of housing in each city, I simply divided it by the median annual salary. This gives us a nice index which tells us roughly how many years you’d have to work in a particular city to be able to afford to buy 100m² worth of property.

Results

Salary

The top 3 cities in my sample by median salary:

1. Geneva, Switzerland – £77,313
2. Zurich, Switzerland – £76,588
3. Tokyo, Japan – £48,728

I was shocked by the gulf between the Swiss salaries and everyone else. However, everything is famously expensive in Switzerland, so taking into account food, utilities, transport and housing, they may be justified such a high salary.

The bottom 3 cities in my sample by median salary:

102. Karachi, Pakistan – £8,579
103. Dhaka, Bangladesh – £7,883
104. Colombo, Sri Lanka – £6,847

I didn’t include many African countries in my sample. Clearly, many of their salaries are likely to be lower.

Housing

The 3 most expensive cities in my sample by average cost of a 100m² property:

1. Hong Kong – £1,302,967.50
2. Tokyo, Japan – £909,432.50
3. Rome, Italy – £768,687.50

I was quite surprised at the fact that Rome beat out London, Paris and Singapore.

The 3 cheapest cities in my sample by average cost of a 100m² property:

102. Karachi, Pakistan – £35,460
103. Lahore, Pakistan – £28,694.50
104. Dhaka, Bangladesh – £18,464.50

Ratio of salary to housing

The 5 cities with the greatest proportional disparity between salary and housing:

1. Hong Kong – 43.98
2. Singapore – 25.55
3. Rome, Italy – 25.45
4. Taipei, Taiwan – 25.13
5. Moscow, Russia – 20.76

This means that, theoretically speaking, it would take you 44 years of working in Hong Kong to buy 100m² worth of property! Of course, this ridiculous figure is the reason why apartments in Hong Kong are so tiny. This is the reason why I would have preferred to get actual median house price data. 100m² is simply not a realistic property area for Hong Kong. Clearly, actual average home prices would be lower to reflect the smaller average size. Nonetheless, it is a rather frightening statistic (if not of living costs, then certainly of the extreme density and lack of space).

The 5 cities with the smallest proportional disparity between salary and housing:

100. Montreal, Canada – 3.55
101. Istanbul, Turkey – 3.25
102. Atlanta, USA – 3.21
103. Ankara, Turkey – 2.83
104. Chicago, USA – 2.69

So it seems North America and Turkey are the way to go if you want to live comfortably!

A final interesting statistic to come out of this was the difference between inner city and outer city housing prices. In almost all cities, the city centre was more expensive to buy property in than the outskirts. However, in 3 cases (Johannesburg, Durban and Santiago), property costs more outside the city centre than inside. One explanation may be that the inner city areas in these places have higher levels of deprivation than the outer city areas, when the opposite may hold in most other places. However, I do not know enough about these places to make an accurate judgement. If you have any thoughts or experience in this area, feel free to leave a comment.

The full dataset can be downloaded here: salary vs house price

Here’s a video of me playing Joe Satriani’s classic – Always With Me, Always With You. Enjoy.

A bit of history

Marcus was a prized disciple of the famously foul-mouthed Gordon Ramsay. Under Ramsay, Marcus opened up Petrus at The Berkeley. It was a very successful restaurant, gaining 2 Michelin stars. However, Marcus and Gordon had a rather high-profile falling out. Because the restaurant was owned by Ramsay’s company, he still owned the rights to the name. Marcus managed to buy out the restaurant in a legal battle, but lost the Petrus name to Ramsay. As a result, he renamed the restaurant after himself and Ramsay opened a new Petrus down the road.

The Environment

As you’d expect from a restaurant situated within a hotel that charges around £250 a night, ‘posh’ and ‘pretentious’ was the atmosphere that you got. It was a great experience, although I did feel a little out of place there, especially when your average hotel resident was walking around casually in a 3 piece suit.

The restaurant itself is actually a modestly sized room. It was nicely decorated, but didn’t feel over the top. The overall atmosphere was cozy and understated. A pleasant place to enjoy your meal; some might expect a little more for the kind of status it holds.

The Service

If the decor was subtle, the service was definitely not. I was escorted to the toilets because they were not so obvious to find. When I came back, my napkin had been replaced and neatly folded. If my glass of water dropped to beneath about 25% full, someone would magically appear from nowhere to top it up. I’m not used to this. At first, I felt compelled to thank them each time, but eventually forced myself not to to save energy and awkwardness. I guess some of the more affluent diners were used to this and so had higher expectations, therefore saying less.

Everyone was delightfully friendly, even when we started asking seemingly unorthodox questions about what some of the terms in the menu meant (speck is ham, apparently), and why they did some of the things they did. I suppose they are paid to behave in this way, but my expectations are usually not that high so I was impressed nonetheless.

The only criticism of this type of attention is that if you’re not used to it, it can seem overbearing, therefore making it difficult to relax. To start off with, there was someone at our table every time I tried to start a conversation. I don’t see why all the top restaurants can’t tone it down at least a little.

Starter

With each of our 3 courses, we were first given a mini ‘experimental’ dish to try out, which was a nice little bonus. First up for us were some sesame ‘crisps’ with an aubergine dip. Nothing special, but they would have been fine if they didn’t contain the salt content of the Dead Sea. After most of the water had left my body via osmosis, we were brought a selection of interestingly flavoured breads along with butter. I love bread, and so I lapped this up. They didn’t quite have that warm and ‘freshly baked’ taste that I’ve experienced before, but it was exceedingly good bread.

For our actual starter, we both ordered the pea ravioli with bits of apricot and ham/bacon. Oh, and pea flowers with a pea sauce. Garnished with peas. Obviously, they had a posher name for it, but I don’t have the foggiest idea what it was. Note to posh restaurants: name your dishes things so that people don’t have to ask what’s in them. Or put in a layman’s translation.

The presentation was nice, but I’m not that fussed about what it looks like. Taste is paramount. I tucked in. Oh look, more Dead Sea salt. I’m sure the ravioli would have been nice but all I could taste was salt, and I believe it came mostly from the sauce. As you can see, the sauce was difficult to avoid. It also had a strange aftertaste to it that I wasn’t too keen on. This was definitely the worst dish of the 3, and I did mention the excessive saltiness to the waiter.

Main Course

The experimental dish before the main came in the form of a cappuccino-like object made from sour cream and other foamed delights. I’d always imagined that I’d only be drinking food when I end up feeding through a straw in hospital, but once again, the world of fine dining breaks down all preconceptions. The texture was nice, being that I enjoy frothed coffee-like beverages, but I’d rather have it have tasted of coffee too, rather than some sour (and honestly, slightly salty!) stuff.

For the main, I plumped for pollock and Mr Fonz chose the chicken. My favourite course. The fish fillet was in a nice neat little cube surrounded with greenish beans and other garnishes. The pollock was cooked perfectly – the best cooked fish I think I’ve eaten. It had a perfectly crispy skin on top. No harsh flavours, no excessive saltiness. A very well prepared dish. Apparently, the chicken was also very well cooked. Thumbs up here then.

Dessert

The experimental dessert dish looked like a strip of toothpaste. Except it was pineapple flavoured ice-cream, and was extremely refreshing. Not much to say here. A nice, but small, treat.

For dessert, both of us chose the custard tart. I can’t remember what the other option was, but I remember it sounding pretty boring, so this one was a no-brainer. As you can see, I’d say this was the nicest looking dish we were faced with:

The custard tart was served with some strawberry (or raspberry) sauce, dollops of cream and strawberry ice-cream. Finally, the green blobs you see were tarragon flavoured ice-cream. Yes, you heard me correctly. The custard tart was very nicely done – it had a gentle flavour and lovely, smooth consistency. For me, the best part of any tart is the pastry, and again, this was done to perfection. Solid enough to hold together and add crispiness, but brittle enough to break and crumble. However, it did seem a bit ‘boring’ for a dessert in this sort of establishment, and considering that this was the better of the choices, it was a little disappointing. I guess that’s what we got for taking the so-called ‘cheap’ lunch option.

The one thing neither of us could stomach was the tarragon ice-cream. Up until this, although I’d had issues with saltiness and some of the flavours, there wasn’t anything that I’d left. I really tried to like it, but in the end, my tongue could take no more. As it melted, I frantically tried to stop it from running into my precious tart with mixed success. Mr Fonz also left his share, and we both shared our puzzlement at this choice of flavour with the waiter. Politely, I exclaimed “I think it’s a bit of an acquired taste”. That it was, but I’d like to see some statistics on what percentage of diners had genuinely ‘acquired’ a taste for it.

Finally, we were brought a selection of freshly-made chocolates. We tried each of the 7 or 8 flavours in a mini tasting competition and concluded that the banana caramel truffle was the unanimous winner. Some of the flavours were quite ridiculously overpowering (fennel?!), and I think this sums up the whole dining experience. Simple and effective ideas were somewhat tarnished by needless pretence (by way of adding strong flavours) where it wasn’t needed.

Oh, and finally finally, we were given the painful bill. I must say though, that by London prices, it wasn’t bad. Also, I appreciate the practice that some of the more exclusive restaurants are employing – to add a 12.5% service charge automatically to the bill. I don’t see why waiters aren’t paid a fixed fee like any other profession. The process of tipping seems haphazard and unfair.

In summary…

Marcus Wareing at The Berkeley turned out to be a mixed bag of delights and, well, salt. It was certainly a unique culinary experience. But I have been to places that are cheaper and have left me more satisfied in the stomach (notable example: Pearl by Jun Tanaka, which I think dominated Marcus Wareing in every way). Sure, a lot of what we paid for were the frills that came with the dining, and they were entertaining. I’m also sure that the reputation and Michelin rating added a premium to the price. For me though, none of the frills are worth it if the core – the food – is not up to par. After all, the food is why I go out to eat in the first place.

To try to balance the argument, I’d considered that maybe the full dinner and a la carte menus had better dishes. From what I read, the dishes did indeed sound more interesting. But, as Mr Fonz was quick to point out, a highly acclaimed restaurant should have the same excellence in taste throughout its inventory.

Marcus Wareing at The Berkeley. A restaurant whose food was preceded by its reputation.

The type of shuffle in the photo above is called a ‘riffle shuffle’. I’m sure you’ve all seen it before. You split the deck into two parts and interlock the parts together by using a fanning motion. The way most people do it in general is quite randomly and haphazardly, which is fair enough because the idea is to mix the cards up as much as possible.

However, you can be a bit more precise. You can perform a ‘perfect’ riffle shuffle by splitting the deck into 2 exact halves (26 cards each) and interlocking the cards so that the cards from the 2 halves alternate into the final pile. To be a little more precise, if we call the two halves A and B, then we want to combine the cards so that every other card in the deck after the shuffle is from pile A.

Now, you may already have realised that there are exactly 2 ways of doing this. Suppose half A is the top half of the unshuffled deck and B is the bottom half. Then you can perform the shuffle with either:

1. The top card from B ending up on the top in the shuffled deck
2. The top card from A ending up on the top in the shuffled deck

Situation 1 is called ‘middle card top’, and 2 is called ‘top card top’, because these are the cards that will end up on the top of the shuffled deck.

It may seem like a trivial point, but magicians will know that this is not so! If we keep repeating the ‘middle card top’ shuffle, then we’d have to do it 52 times to return the deck to its original order. However, if we repeated the ‘top card top’ shuffle, we’d only have to repeat it 8 times to get back to the starting deck! This seems a little surprising!

Can we prove formally that this will happen every time without physically shuffling the cards and making a mess? Of course we can!

Middle card top

Let’s give each card in our unshuffled deck a number from 1-52 i.e. the top card is numbered 1, the second 2 and so on.

After a middle card top shuffle, the 27th card will be the top card in the shuffled deck. The 28th card will be in position 3, the 29th in position 5 etc. Therefore, the ordering of the shuffled deck after one iteration is:

27, 1, 28, 2, 29, 3, 30, 4, 31, 5, 32, 6, 33, 7, 34, 8, 35, 9, 36, 10, 37, 11, 38, 12, 39, 13, 40, 14, 41, 15, 42, 16, 43, 17, 44, 18, 45, 19, 46, 20, 47, 21, 48, 22, 49, 23, 50, 24, 51, 25, 52, 26.

The clever bit is that we don’t have to repeat this 52 times! We can see how the cards will cycle after shuffles by noticing that the 1st card becomes the 27th. The 27th card then becomes the 40th according to the ordering above. If we follow the above chain, we can write down the cycle in the following form:

(1, 27, 40, 20, 10, 5, 29, 41, 47, 50, 25, 39, 46, 23, 38, 19, 36, 18, 9, 31, 42, 21, 37, 45, 49, 51, 52, 26, 13, 33, 43, 48, 24, 12, 6, 3, 28, 14, 7, 30, 15, 34, 17, 35, 44, 22, 11, 32, 16, 8, 4, 2)

What does this tell us?

The first thing you should notice is that we have one big chain of numbers that loops around. The length of this loop is 52. This means that to get card 1 back in the top position, it has to travel through each of the other positions in the cycle. In other words, we have to repeat the shuffle exactly 52 times to get each of the cards back to their starting positions.

Now that we know what to do, it should be easier to look at the second shuffle.

Top card top

After one iteration of this shuffle, we get the ordering:

1, 27, 2, 28, 3, 29, 4, 30, 5, 31, 6, 32, 7, 33, 8, 34, 9, 35, 10, 36, 11, 37, 12, 38, 13, 39, 14, 40, 15, 41, 16, 42, 17, 43, 18, 44, 19, 45, 20, 46, 21, 47, 22, 48, 23, 49, 24, 50, 25, 51, 26, 52.

Writing this in cycle form, we can immediately see that there will be multiple cycles, because the first card always stays on top. Similarly, card 52 is still card 52 after an iteration of the shuffle. Writing out all of the cycles gives:

(1) (2,27,14,33,17,9,5,3) (4,28,40,46,49,25,13,7) (6,29,15,8,30,41,21,11) (10,31,16,34,43,22,37,19) (12,32,42,47,24,38,45,23) (18,35) (20,36,44,48,50,51,26,39) (52)

We can see that there are two cycles of length 1, one cycle of length 2 and six cycles of length 8. The lowest common multiple of 1, 2 and 8 is 8. Therefore, to reach the starting position, we only need to perform 8 shuffles of the top card top kind.

Remember folks, dexterity and maths are the key to becoming a successful close-up card trick expert!

First things first – let me say that LA Noire is an enjoyable experience with a gripping plot. I had a lot of fun playing through it. That said, as a piece of interactive gaming, it falls short in a couple of key areas.

Open world?

LA Noire is a Rockstar game at heart. It uses the same engine and open-world mechanic that made the Grand Theft Auto games so rich and lasting. But you have to understand why GTA games ‘work’. Sure, they have a vast map area to explore, but the key to longevity and fun comes from the missions and objectives that it has you do. During the course of a GTA game, you work for different people located around different areas of the map. Each phase of the game helps you get familiar with a part of the map. Other ‘side-missions’ are designed well too – again, because they take advantage of your environment.

Like GTA, LA Noire has a huge map to explore. But the point is, none of the cases you have to solve involve exploring any of it. You just go from crime scene to various locations until you close up the case. Yes, there are things like hidden cars to find if you so wish, thereby allowing you to explore the map further, but why should you bother? Unless you’re someone who likes to tick everything off a list, there is absolutely no value in the game to finding anything ‘hidden’, in that you gain no advantage by doing so.

Driving from location to location is fun at first. You soak up all that 1940s LA atmosphere and enjoy the erratic yet sluggish handling of the classic vehicles. But very, very quickly, it gets boring and I found myself skipping all the journeys after not very long (thankfully it gives you the option to do so).

All of this leads me back to the title of this article. Heavy Rain – the PS3 exclusive released last year – was a revolutionary mystery solving interactive movie-like experience. It based its gameplay on a variety of context-sensitive QTEs (quick-time events). You know: press the correct button when it flashes on screen and sometimes choose from different actions/responses to situations.

Heavy Rain’s gameplay had its criticisms too, but it knew what it was. The game was totally linear in that it popped you in different scenes according to the story. The problem with LA Noire is that, at its core, it is a mystery solving, linear interactive story just as Heavy Rain was. But rather than accept this fact and optimise the gameplay experience to focus just on the things you can do in each scene, they had to give you the illusion of an open world sandbox type game, which it is most certainly not. As a result of this, the crime solving, clue gathering scenes which make up most of the game are so one-dimensional in gameplay terms that it is unbelievable. Walk around a room or area, wait for the controller to vibrate and press X. Rinse and repeat until the music stops so you know you’ve found all the clues. Then move on to the next area.

The compelling plot is what holds the thing together, but without that, the game would be a joke. Also, most of the locales you visit (especially in the homicide portion of the game) are so similar that it’s unbelievable. If Team Bondi had canned the whole open map and spent that development time on making each scene unique and fun to play, the game might just have been a classic.

Interrogation

Other than clue finding, the other main game mechanic in LA Noire is performing interrogations on suspects and witnesses. The big draw to this feature, and indeed, the thing that makes it all possible is the amazing facial motion capture system. For the first time in a game, faces look believable and lifelike. The acting is also excellent, and this helps to immerse you in the plot and characters further. LA Noire has definitely raised the bar on facial animation, and I hope that more story-driven games will use the technology in future.

With that said, it is unfortunate that the interrogations are repetitive and give you the same three options each time: Truth, Doubt and Lie. The idea is that the subject can either tell the truth or lie/withhold information. If you have proof they are lying, you select ‘Lie’ and choose the piece of evidence that is relevant. If you have no proof but don’t trust them, you’d select ‘Doubt’.

Unfortunately, the 3 options are limiting. Shifty eyes mean that the subject is probably lying, and apart from the odd few difficult suspects, it’s usually fairly easy to tell. However, this doesn’t mean you’ll get it right! As Yahtzee explains in his excellent Zero Punctuation review of the game, the problem is that the options are completely ambiguous. Sometimes you know that the guy is lying and have loads of evidence against him, but selecting the wrong one will have the game shoot you down like the loser it thinks you are. Other times, you might feel that part of their statement is inaccurate so you choose Doubt, only to have your character go completely schizophrenic with accusations. In reality, there could be more than one right answer, but the game is quick to make it clear that there is only one correct path. Not that it makes a whole lot of difference if you get things wrong anyway – the game still progresses quite normally.

This is another area where LA Noire could have learnt from Heavy Rain. I appreciate that it may be more difficult from a development point of view, but having situation specific responses a la those found in RPGs would really have made interrogation more interesting and given the acting a real chance to shine through. If they’d removed all the peripheral rubbish and stuck to making the core gameplay a bit better, it may not have been much of an extra development burden.

They might also have been able to make the game longer. At about 12 or 13 hours, it really doesn’t compare to other Rockstar games that use the same open world mechanic. But it is at heart a different game to GTA IV or Red Dead Redemption, and so they shouldn’t really be compared. That’s why I think that they should have approached the game more like Heavy Rain, and less like a cookie-cutter Rockstar title (hell, they already stole a trademark plot device from it). Maybe it would have elevated LA Noire from a decent game to an all-time great.

Maths is full of cool theorems. The Pigeonhole Principle is one of my favourites because of the fact that it is so powerful and quite tricky to prove, yet so intuitive and easy to understand. Before I can talk about it though, I need to introduce you to the world of mathematical functions.

Functions

In maths, a function is a ‘mapping’ from one ‘space’ to another. The ones that people are used to seeing are things like f(x) = x² but functions don’t have to involve algebra at all. I find that it’s easiest to explain them using diagrams and by giving real names to things, so here goes…

Have a look at the diagram above. There are 2 spaces or sets, which in this case are the countries England and Australia. England is the domain – the place that we are mapping from, whilst Australia is the co-domain (also called range), and it is the place that we are mapping to. England and Australia both contain cities, which are represented by the nodes within each set.

In this scenario, a function is a set of roads that takes us from somewhere in England to somewhere in Australia. All the English cities must take us somewhere – so a set of roads is a valid function only if each city in England has a road going to some Australian city. The roads are one-way only in our case, so sadly no Aussies can return to the homeland.

There are a few types of road arrangement that have special names:

Injection

An injection or ‘injective function’ is when each element in the co-domain is mapped to by at most one of the elements in the domain. In our example, to have an injection, we are not allowed to have an Australian city that has 2 (or more) roads coming into it from different English cities. We couldn’t have roads from both London and Manchester going to Sydney, for example.

The diagram below shows one possible injective function. Sydney, Canberra and Perth each have one road coming into them from distinct English cities. Melbourne has no roads coming into it, but that’s fine for an injection because nobody really cares about Melbourne anyway…

Surjection

A surjection or ‘surjective function’ is when every element in the co-domain is mapped to. So now we need to show love to all the Australian cities – even Melbourne. Each city in Australia must have a road coming into it to have a surjection. Importantly, they don’t all have to come from unique cities in England. As you can see below, all 3 Australian cities are mapped to, but notice that London has 2 roads coming out of it whilst Sydney has 2 going into it. This wouldn’t be allowed in an injection, but it is okay for a surjection.

Bijection

Finally, if a function is injective and surjective, then it is bijective. In other words, each English city must have one road coming out of it and each Australian city must have one road going into it. Each English city is ‘paired’ with an Australian one.

Notice a key feature of the bijection: you cannot have a bijection when there are different numbers of cities in each country. If you look back at the injection example where we had Canberra as well, there is no way to pair all of the cities together between the two countries.

In high school, I remember thinking how our grading system was a total farce. We weren’t graded on ‘results’, allegedly. Instead, our teachers gave us a grading based on our ‘effort’. There were 5 grades in total:

C – Commendable
G – Good
S – Satisfactory
D – Disappointing
U – Unsatisfactory

To me this was complete nonsense, because effort is not observable! What the teachers did was to look at the end product, and use that to infer some ‘perceived’ level of effort to base their grading upon. This isn’t the same thing as actually judging effort. I knew this because, quite often, my end product was pretty good and although it required minimal effort, I ended up with the top grade of C. Similarly, there were times where I put in a lot of effort, but got a poor effort grade because my end product was not so good.

Asymmetric Information

This situation is a classic example of ‘asymmetric information’. This is where one person (or party) can do some action, or has some information, that the other person cannot observe fully.

In terms of effort, this means that students could get away with putting in much less effort because teachers cannot accurately observe or measure it. On the flip side, teachers may misread a student’s effort based on their work.

My effort

I’m going to construct a fairly trivial model in the context of my high school effort grades to make things a little bit interesting.

It is true that the economy is suffering from unemployment and reduced consumer confidence even now, which is why budget cuts and tax rises are arguably not helping matters. This is likely to be the reason that the Bank of England, up until now, have decided to keep interest rates at the lowly 0.5% for well over a year.

I was planning to predict an interest rate rise sometime this year and I might have sounded a little smarter if I wrote this article when I intended to in January! Now, it seems that people are all but certain that rates will go up at some point this year. Why is this?

Inflation!

Ahh yes, inflation. The magic word in monetary policy. The retail price index measure of inflation was announced as being at 5.5% p.a. and the consumer price index was announced as being at 4.4% p.a today. That is some way over the 2% target set by the government.

Food prices have been noticeably higher in recent months, but so have many other regular items of expenditure – from fuel to clothing.

This seems to confirm the fact that the original Phillips Curve relationship, which stated that there was a trade-off between unemployment and inflation, was really a phenomenon of the past (or a short run effect at best). This was also evident in the boom phase of the early 2000s, which saw low inflation and low unemployment – the polar opposite of what we are experiencing now.

According to the Taylor rule of monetary policy and the Fisher equation, optimal policy suggests a rise in interest rates higher than the increase in inflation is necessary to bring the real interest rate down, creating disinflationary pressure. So why have the Monetary Policy Committee been so cautious to raise interest rates?

Delicate economy

The obvious concern is the fact that the economy is still suffering the effects of the recession – possibly more so now in the so called ‘recovery’ phase. The risk is that with higher interest rates, it will put an even tighter squeeze on already suffering households by increasing lean and mortgage repayments and hence discouraging consumption. With the recent announcement that consumer confidence has fallen, raising interest rates is certainly risky.

Expectations and Credibility

Something that may go unnoticed by the average person is that monetary policy can actually happen just because people expect the Bank of England to do something. Because it is well known that they have a duty to maintain a 2% target level of inflation, when inflation gets high, people expect interest rates to go up. When this happens, they start adjusting their behaviour before anything actually happens.

Source: www.bloomberg.com