# Subjective Complexity

Complexity here, there and everywhere. The term complexity is omnipresent. Questions are complex, as are problems, solutions, systems, projects, ventures, requirements, markets and so on. But is this really true or is it just a subjective perception. Is it worthwhile to try and calculate complexity, and how important are the results for your individual perception of it?

## Complexity, noun, [kəm-ˈplek-sə-tē, käm-]

Maybe the shortest definition of the term complexity is *the state or quality of being complex; intricacy.*¹ What does *intricacy* actually mean? How many layers, factors, features turn things into complex ones? For example, some DVD players are said to be complex and difficult to operate, others are said to be be less complex. I am sure you are well able to play a DVD on any player but what about other members of your family? The perception of complexity is a personal thing.

## The Personal Perception of Complexity

Wanna play a game? A simple but revealing one. Come and join me!

You see an equation made of matches, 2 + 2 = 5. Based on what we know about mathematics, this is wrong. Change just one match and it is correct:

Do you have the solution? It is, of course, 3 + 2 = 5! Piece of cake. Basic arithmetic operations is knowledge we can instantly access. But finding the solution also requires visual skills. Which one is the match that must be moved? Maybe you tried changing the equation to 2 + 2= 4. This would also be correct, but it is also impossible by changing just one match.

Ok. Let us not try to create suspense where there is none. There happens to be another solution, a different match that can be moved, 2 + 3 = 5. Did you try to find this alternative solution after finding the first one?

Let us continue our quest for a personal perception of complexity. A new equation: 4 x 8 = 2. Two matches must be moved in this case. Find all possible solutions.

Moving two matches is more difficult, isn’t it? You’re gonna be fine.

4 x 2 = 8. The first solution. Does it make a difference to you that you know there is another solution?

4 x 3 = 12. Things are a little different here. One number must be altered and another one must be added. The task may be described as something like *Adjust the equation by moving two matches while only altering one number.* Sounds complex, doesn’t it?

4 x 0 = 0. Ok, we get the idea. Finding solutions gets easier and easier. *All beginnings are difficult*, as they say.

1 x 8 = 8. The fourth solution. Also, the final one.

How difficult was it for you? Well, this was only a game, no external pressure, no audience. What difference would it have made if you knew there are four solutions? Would it have affected your perception of complexity? Is there a number of solutions that would scare you away?

There is no way to determine general rules for the perception of complexity; people used to mental excercises probably do not regard this game as being complex. Once the main principle is clear the solutions come to you. Whether you regard a task, a question, a game or a system as complex depends on you, and only you:

- Your understanding – have you understood the task in its entirety?
- Your skills – altering an equation is impossible without knowledge on basic arithmetic operations.
- Your experience – are you already acquainted with the task?

## Calculating Complexity

Maybe you will be surprised to learn that complexity can indeed be measured and calculated. The term *variety *is commonly used as a measure of complexity. Usually, the term refers to *diversity *of some sort; the problem here is the diversity of possible interpretations. In anatomy, variety refers to a deviation from the norm regarding, for example, size or form of organs; in mineralogy, it refers to a deviation regarding build of minerals, e.g. colour, size or habitus.

Let us now take a closer look at the calculation of complexity in simple, non-complex systems: a torch. Usually, a torch has two states, *on* and *off*. How many states are there in total if we have five torches? Combinatorics tell us that the answer is *2 ^{5 }= 32.* This way, complexity becomes a real, objective parameter. How many states are there if we have 25 torches?

*2*.

^{25 }= 33.554.432Impressive, but of what use is all this? Does it have an effect on you to know the exact number of possible moves in chess playing? There are 32 chess pieces and the number of possible actions is large, some sources say there are 10^{155 }which would result in a 10 with 155 zeros. Chess seems to be complex; whether this is a problem for you depends only on your perception of it.

## Conclusion

The amount of complexity increases with the number of features. There may be an objective, calculable degree of complexity but it is up to you to evaluate and judge it. A torch with two states is not complex; at least it is not complex to operate. If you take a look at the construction of a torch the situation looks quite different. What counts here is the point of view. Complexity is subjective.

P.S.: I will play the above game with the six-year old daughter of a friend; I wonder how difficult it will be for her.

Notes: [1] Definition of complexity at Dictionary.com, last retrieved September 1, 2015

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