Hitting the Intellectual Wall

Athletes have their 'Wall': a psychological and physiological barrier that requires considerable effort to break through. Not being a particularly sporty type, I had not really thought that it might apply to other endeavours. But the recent emergence of a pattern (I'm definitely an 'emerging pattern' type of person - which is an interesting subject all to itself!) that I'd sort of glossed over made me realise that the concept might apply to a much wider scope than I'd thought.

The trigger was something I dabble with in rare idle moments: Sudoku. I was first introduced to it at a Wedding reception where they handed out sheets of brain teasers to occupy the guests whilst all the wedding reception stuff was being prepared, and there were two Sudoku taken from a broadsheet newspaper. Never having seen them before I solved them and thought that they were quite diverting. Since then I've got one of those electronic ones (naturally), and another one that has possibly the worst cursor-moving device ever perpetrated on human-kind, and several computer, iPod and iPhone versions.

The problem is, there's a sub-set of Sudoku problems that I can't solve. I reach a position where all of the remaining squares can be filled by one of a pair of numbers, and I can't figure out a way of proceeding past this block. I've reached my own 'Wall'. I've spent quite a while analysing the ones that I can't do to try and see if there's something I'm missing, but I haven't found it yet. Here's one:


Whilst I was racking my brain trying to solve it, I remembered a past situation with a close resonance to this one: when I tried to learn the 'Lisp' programming language. Now every other programming language I've learnt involved an initial feeling of being overwhelmed by the arcane difference between the newcomer and what I already knew, but eventually this was replaced with a moment of inspiration where 'I got it' and then things went as smoothly as learning a new language ever goes. This was how it was for Fortan, Basic, C, 68000 assembler, Pascal, HTML and Prolog...

But Lisp was different. Big Time different. 'You aren't going to like this' different. I unexpectedly hit the Wall and could not get past. And did I try! There was something there that just didn't compute. Just like this sub-set of Sudoku problems. Now I'm not saying that this isn't solvable, merely that I can't figure it out. Quite humbling, really.

So in my case, my own personal Wall is very tangible. Maybe I just need to keep metaphorically running, but it is certainly fascinating to see one close up and personal.

Deep waves

One of the hardest problems in designing something is deciding where the 'sweet spot' is - just enough facilities to provide lots of possibilities, but not too much because that wastes money. Some current Digital Audio Workstations (DAWs) provide lots of dedicated plug-ins for a variety of synthesis techniques, whilst others go for more novel approaches.

Abelton Live is one of the latter, with only a few dedicated virtual synthesis plug-ins. But by combining the simple sampler playback (via the brilliantly named 'Simpler') with the built-in MIDI and audio effects processing, and the general purpose 'Rack' that lets you build up your own special-purpose 'super-plug-in', you can do some amazing things.

This was highlighted for me with two recent purchases. I bought 'Lumen' for my Apple iPhone 3G, and 'Vector' from Puremagnetik.com for my Ableton Live 7 rig. Lumen is a puzzle game where you solve the problem of passing a beam of correctly coloured light through a series of checkpoints, using just a single light source, some coloured filters, and some mirrors. Rather like the Honda 'Cog' advert, both are variations on what is known as a Heath Robinson machine in the UK, or a Rube Goldberg machine in the USA, and they go by other names elsewhere. The machines build complex operations from simple parts, although practicality is not always high on the agenda.

The 'Vector' instrument rack produces sounds reminiscent of one of my favourite pieces of hardware: a Korg Wavestation. The Wavestation uses two distinctly different but related types of synthesis technique: vector synthesis and wave-sequencing.

• Vector synthesis was first introduced in the Sequential (Circuits) Prophet VS, and enabled a dynamic mix of up to four separate ROM sample-playback sounds either using a joystick controller, or stored as a 2D controller timeline.
  
  
• Wave-sequencing takes ROM-playback of samples, and allows the samples to be cross-faded together in a time sequence. This provides sounds with varying timbre, or a rhythmic character.

The 'Vector' instrument rack does not use a dedicated synthesiser, instead it uses the Simpler sample replay instruments, and puts them into time-sequences by using a series of built-in 'Note Length' MIDI processors to produce the timed note triggers, and it uses the ability to map parameters deep inside an instrument rack to be controlled from a simple top-level set of controllers. Having spent many a happy hour creating wave sequences, I was intrigued to discover that the process is very similar in the original hardware and this modern virtual form.

In fact, the virtual form might even be faster for making sequences, and it has the added bonus that the wave sequencing is not fixed, and can be edited with all of the MIDI and audio processing, plus all the instrument rack features like zones, velocity switching etc., plus the samples are no longer locked in a Korg ROM!

Being able to make and edit a synthesis technique by bolting together simple processors and instruments is a wonderful thing! For me, it is more satifying than solving a puzzle like the thousands inside Lumen... but it takes rather more time, and so far, isn't as portable!

A Question of Sides...

Yesterday's thinking about preconceptions of 3D reminded me of something that I was shown as a child, and which I now know to be a classic puzzle. But armed with 3D thinking, then there's a couple of wrinkles that I'd like to add to the canon...

The premise is simple enough. There are three houses, and three utility suppliers: Electricity, Gas and Water. All you have to do is connect the three houses to each of the utilities, with the one constraint being that you cannot have any of the utility supplies crossing over another - some obscure local bye-law or something is usually part of the patter at this point. The diagram that accompanies the problem is straight-forward:

The start position

People often try connecting one utility to all three houses, and then the second utility, and then the third, two of which are normally easy, but the third is blocked by a previous connection. Back-tracking and re-routing only results in the same blocking, and persistence or ingenuity are no help in finding a solution.

Only the water connection to the middle house to go...

Actually, there isn't a solution, and the mathematics behind it is very well understood, and was neatly captured by Euler in a formula that describes the properties of 'planar graphs' and 'non-planar graphs', of which this is an example. It actually boils down to a planar graph being a collection of points and lines connecting them that can be drawn on a surface without any cross-overs, whilst a non-planar graph is one where you can't. The three houses problem is non-planar, and so can't be solved on a flat surface. Of course, now that you know that you can try to solve it on other surfaces, and a doughnut-shape (or, more formally, a torus) is one example of a surface on which you can solve it.

But one surface which most people forget about is the other side of the piece of paper on which this problem is normally drawn. By using the other side of the piece of paper, it is also possible to solve it. However, one of the consequences of technology is that we have new ways of representing things, and if you draw the diagram on a screen, then an interesting thing happens - you can't use the other side, because:

'Unlike paper, screens don't have an 'other' side.'

Which made me sit down and think hard, because I'd never thought about it before. Screens are single sided, rather like a möebius strip.

If you want to know more about the history and the maths behind this puzzle, then there's an excellent web-page here.