In a recently published course (Tao, 2021), Terence Tao, a maths prodigy, shared some of his insights into problem-solving for real-world applications. Tao had a keen interest in mathematics and earned his PhD from Princeton University when he was just 20 years old. He obtained his professorship at age 24 and has been able to apply his understanding of mathematics to deliver groundbreaking innovations in healthcare, particularly in making magnetic resonance imaging (MRI) scanning faster and more efficient.
This article is not about applying complex maths to everyday issues—the goal is to gain a high-level understanding of some of the strategies used that can help in both marketing and clinical research.
Asking the right question
Sometimes, to solve a problem, we need to change our perception, make sure that we are asking the right question in the first case and try to avoid our thinking getting stuck in a rut. For example, an airport was getting numerous complaints from passengers about the length of time that they had to wait in the arrivals hall to get their luggage back. Despite the airport management implementing numerous improvements to baggage handling to deliver small incremental improvements, the number of complaints stayed more or less the same.
Upon further examination of the problem, the passengers did not like the amount of time they spent waiting in the arrivals hall, not the overall time it took for the baggage to be unloaded from the plane to be processed and moved to the baggage carousel. The solution was simple and very effective: make the passengers walk a longer distance by using partitions to make them take more time to get to the arrivals hall, thereby reducing their waiting time. In this case, the solution was to understand the problem in more detail, which, in turn, changed the question being asked.
Solve the problem with a story
Applying a narrative to a problem by adding a real-world context can give new insights into how to solve it. The key to building up these stories is to ask quite ‘dumb’ questions about the problem and build up our intuition.
For example, if you multiply two negative numbers (for example, -4 × -3), you end up with a positive result (12), which is not intuitive to a layperson. If we apply an economic analogy to help, we can make sense of this. If you assume you work for 3 hours and get £4 per hour, you will get £12. However, if it costs £4 an hour to do an activity, such as parking your car in an expensive car park for 3 hours, parking there will cost £12. If we decide to park somewhere else for free, such as a side street, for example, we will save £12 instead of generating it.
Applying a real-world context to an abstract problem has merit when thinking about finance and process design. It makes the problem more relatable to others and enables you to approach the problem from a different angle.
The group testing problem
Another technique is to break complex problems into smaller components that are more manageable. In the 1950s, the US army needed to test large groups of patients for a rare disease. The problem was that the tests had a very limited supply, and the condition had very low prevalence, meaning that very few patients were expected to have it.
Rather than test every single blood sample from 100 patients, a method known as a binary division protocol was used. A small amount of all the blood collected from the patients was mixed together and tested to see if anyone had the disease. If the test came back positive, the patient results group was split into two groups of 50 and tested again. Each time a positive result was found, the positive group was divided in half again until the single positive sample was eventually found.
The interesting thing about using this method is that, to find a single positive result out of 100 samples, you only need to divide the groups seven times and use no more than 13 tests on average. This is a considerable improvement on running the test 100 times to find no result or a single positive result and has significant implications for public health monitoring efficiency.
Understanding polling
Polling is a common way of understanding what a population thinks of a particular topic, such as voting intentions or if they like a particular product and—if done well—can be quite accurate. It is not the proportion of the population that are polled that determines its accuracy, it is the sample size and if that sample is representative of the population as a whole. For example, a representative sample of 1000 people over a population of 60 million, rather than asking 100 000 people in a single geographical area for understanding a national issue. Understanding this is hard, though, so let us apply a good analogy to explain how this works.
» Polling is a common way of understanding what a population thinks of a particular topic, such as voting intentions or if they like a particular product and—if done well—can be quite accurate. It is not the proportion of the population that are polled that determines its accuracy, it is the sample size and if that sample is representative of the population as a whole «
If you wanted to determine the salt content of the oceans, taking a single drop of water from it and tasting it would let you know that the entire ocean is full of salt. The reason we know this is that the salt is quite well distributed in the ocean. However, some of the ocean contains freshwater in addition to saltwater where rivers flow into it. If that one sample was carried out near the mouth of a river, one might decide that the oceans are freshwater instead. In this case, the sample size was too small, and it needed to be randomly picked. It is important that random sampling is representative of the ocean that you are trying to represent.
We used an analogy to understand polling, sample sizes, bias and representational fit. This method allows us to use both the rigorous part of our brain and the intuitive part to understand the concept better. Of course, polling can be used to understand our client base better and using the ocean analogy helps us think about what pitfalls can affect the accuracy of its results.
Managing error tolerance and failure
Traditionally, mathematics has been an exact science. Sometimes, an exact answer to a problem is not necessary, so long as the margin of error can be managed. In some situations, our error tolerance is very low, such as drug administration, and, in other situations, a margin of error is okay, for example, using your phone to navigate to a destination.
When looking at a problem, it is worth defining your error tolerance. However, also keep an eye on that tolerance to ensure it does not drift too far away. For example, letting your phone navigate you to the street your destination is on might be okay in some cases, but what if that street is several miles long?
Accept that failure is a part of learning when it comes to developing a solution. It might be possible to use a simplified model to test your hypothesis in some cases. Sometimes, you cannot solve the problem—try taking a break from it and then come back with a different approach.
Use different visualisation methods
There are usually different ways of thinking about a problem and changing the way we view it will activate different parts of our brain to solve it. For example, graphing data activates the spatial portion of your brain and may help you spot new relationships. Sometimes, talking to someone else about the problem and verbalising it will help gain new insights. In other cases, creating a physical representation of the problem can help, too.
Conclusions
Framing problems with a good narrative or an analogy can make a huge difference in understanding it. When setting narratives, watch out for poor ones that lead you to the wrong solution or to a too precise answer.