How do mathematicians model real-life problems?

Let's take a closer look at the six movie episodes presented. Have you ever wondered how mathematicians can do it? There is, in fact, a particular way to think in mathematics that allows you to solve problems. This is exposed in the article how mathematicians think; therefore, it is not surprising that there is a general procedure in which mathematicians go about solving real-life problems, which can be applied very effectively in everyday life. You can have a go at solving real-life, never-formally-solved-before problems, at our website, in the Mathemagic! section.

The secret revealed!
Below is a general list of steps that mathematicians follow in solving problems:

• Define the problem. Make all terms in the problem precise. What is our objective? For example:
  The fashion episode: We can start by thinking: “what is a knot?” What makes a knot “a tie?”
  The AI doctor episode: What exactly do doctors do? Which part of the doctor's job can be done by computers?
  The love episode: So we are interested in finding the best partner. But what measures can we take to say that one partner is better than the other? In other words, how do we define “best?”

• What makes the problem “a problem?” In other words, what are the constraints imposed?
  The shopping episode: If we stop and select a restaurant, we might miss out on better ones down the line. In other words, we don't know if we are getting the “best” restaurant possible.
  The traffic jam episode: If we change lanes, would we necessarily arrive home earlier?

• Spotting the uniqueness. This sometimes simply comes back to defining the problem in a very precise way. For example:
  The banking episode: We can ask ourselves: how are illegal transactions different from the rest? This allow us to pick up the uniqueness of the illegal transactions, and thereby form an effective solution (which utilizes the technique of outlier detection and clustering).
  The tie episode: We note that to make a tie, we need to perform a certain sequence. So all of our ties need to end in this sequence – and this is an additional constraint of the problem.

• Solve the problem. At this stage, we should know the problem quite well. In mathematics, the hardest part is to define the problem in a precise way. Once defined, it is a matter of applying known mathematical techniques to arrive at a solution.

• Refine your solution. Usually the first model has problems. That is why in software engineering, we keep on having different versions of a program! Refinements in a model usually come in the form of additional constraints:
  The tie episode: we want the knots to be aesthetic: symmetric and not too big.
  The love episode: we want to make sure our current partner is happy with the model's output. (Just joking…)
  The banking episode: We might want to extend the applications to detect other types of fraudulence, for example, to detect virus and spyware in our computer.

Ok! Now it is time to apply the theory! Here are some great tips:

• To start some modeling problem of your own, check out our Mathemagic! section, a collection of irresistible problems for you to discover, apply, and amaze your friends!
• Watch the episodes again; this time, think about how these principles were used.
• Think about all of these principles next time you have a problem to solve, whether it is mathematical or not! Onlt then, can we truly see the power of mathematics in our life.
• Check out books on how mathematicians solve problems. We would recommend the book by George Polya, titled How to Solve It. The book provides excellent insight into how mathematical thinking can be applied to academic settings.

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