Lesson 2.05: Systematic Listing

In 6.03 you are going to explore an incredibly useful problem-solving strategy: making systematic lists. This technique is not only powerful but also versatile, helping you tackle a wide range of problems by organizing information in a clear and structured way.

Imagine you are trying to solve a complex puzzle or figure out all possible combinations of a set of items. It can be overwhelming to keep track of everything in your head. This is where systematic lists come in handy. By breaking down the problem into smaller, manageable parts and listing out all possibilities, you can ensure that you don't miss any important details and can more easily identify patterns or solutions.

By the end of this lesson, you will have a new tool in your problem-solving toolkit that you can use in math, science, and even everyday life.

The art of the systematic list is to avoid writing the whole thing out, while still ensuring that nothing could have escaped. As such it is always accompanied by the meta-task of 'rooting for escapees': trying to imaginatively identify or locate where a case could have been missed.

Systematic listing requires that you have a plan before you begin; often that plan is to change the components of your list one at a time.

Key Terms

Permutations

An arrangement of objects where the order of them is significant

Combinations

A collection or selection of objects where the order of them is not significant

Limit Case

An example that lies at the boundary of possibility according to defined conditions

You are not expected to become expert on the permutations and combinations (a mathematical area known as combinatorics), but it is important to see why such orderly listing is powerful.

Lists can quickly become unmanageable.

There are two particular types of problem that emerge from the world of combinatorics: those that involve using lists and shortcuts to find the number of cases that fit a certain requirement and those that involve using lists and shortcuts to find a limit case.

A limit case is one that lies at a boundary defined in the question, such as 'the latest date on which this will occur', 'the first word with no vowels in', 'the next time that all digits from 1 to 10 will be showing'. The sample question in search of RATES involved both aspects. The question below involves the use of systematic listing to find a limit.

When to use it

Systematic listing is a strategy that caters for problems in which it is easy to generate some solutions, but difficult to find all the solutions. The type of problem must generate options (from a finite list), which can be combined in a number of ways. Such problems can often be visualized as occupying a number of blank boxes which need to be filled - with letters, numbers, or options:

The problems become manageable by considering what goes in each of the boxes independently, and then multiplying the number of alternatives that could occupy each box together. When considering how many ways people can be chosen from a list of 10, for instance, there are 10 alternatives for the first box, 9 for the second, 8 for the third, 7 for the fourth and 6 for the fifth.

The total number of possible ways is 10 x 9 x 8 x 7 x 6 = 30240.

How to use it well

When making this decision it is useful to remember the decreasing multiplication rule: if you are filling a series of boxes and there are 'rí choices for the first box, then the number of choices in the first two boxes will be (n) * (n - 1), in the first three boxes (n) x (n - 1) X (n - 2), and so on. This produces the factorial formula given below the RATES sample question on pages 101-2: if you have four items to arrange in four boxes, there will be 4 x 3 x 2 x l = 24 ways of doing it.
Do write out your list formally.

As with many of these strategies, you are aiming to free up the skeptical, creative part of your intellect so that it can criticize your proposed solution and check it for potential errors. This is possible only if your listing process leaves a physical pattern on the page.

Sample questions

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