## Representations – Black Boxes

The idea of a “black box” is common in engineering, but the idea has much wider application.  It’s a simple idea, really.  Just imagine something that has an inside that is inaccessible from the outside.   Easy, right?  For example, a building with doors and windows that are locked.  Or an iPod, with buttons and a screen, but no way to open it up.  Or a Coke machine, with a slot for money, buttons for selection, and a tray for picking up the can of Coke: you aren’t supposed to just be able to get inside and get the drinks or the money out.  Or a poker player: you can watch her face, and watch her moves over the course of a night, but you can’t see the cards she holds, only the cards she’s played.

A black box is rarely interesting unless its behavior is interesting.  A Coke machine has interesting behavior: we put money in the slot and get a drink out.  We may not know for a fact how a particular Coke machine works, but we tend to have built a model for how the machine works.  The model may be accurate or not – how could we find out?

A model for what’s in a black box need not perfectly match what is actually in the black box for the model to be useful.  The act of coming up with a model for what’s in a black box is often referred to as “reverse engineering”.  This phrase pertains to one of the real-life uses of a black box: in this scenario, a competitor has come out with an interesting gadget and you feel the need to make something at least as good – but the competitor isn’t exactly sharing blueprints with you, and there is only so much you can see from breaking one apart and looking inside.   What a physicist does when looking at the motion of planets and coming up with a model for gravity is similar: you look at behavior and come up with a model for what drives that behavior.  In mathematics, you don’t see much talk about black boxes and reverse engineering, but we do things that fit the same pattern.  For example, when we ask for the formula or expression that matches the numbers in a table, we’re really asking for a model that drives the behavior of a box that takes numbers as inputs and then spits out other numbers as output.

The question “What’s a formula that matches the following table:”

isn’t all that different from the question “What’s a model for a black box:

where inputs 3, 4, 7, 10, 11 will produce outputs 5, 7, 13, 19, 21, respectively?”

The notion of a black box is a simple and accessible framework for a fair amount of interesting math, and I intend to show some of that in later posts.

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