Problem Solving. Be Systematic

Many formal structures can help us to systematically record information, such as lists, tables, charts, Venn diagrams, 2-way diagrams and tree diagrams.

Systematic recording makes it easy to review what has already been done, to easily spot missing or duplicate values, and to identify patterns (which might then be extended).

Worked Examples

First we will read both examples and have a quick think about them and then we will look at how working systematically can help us with each one:

South Korea Example

Below is the South Korean flag:

The trigrams around the outside of the central circle each are made up of three lines. The top left one has solid lines. The one on the top right has broken lines at the bottom an the top and a solid line in the middle.

Using only solid or broken lines, what fraction of all the possible trigrams appear on the South Korean flag.

Postman Example

The postman notice that Suzie had a very large number of cards and packages one day, so guessed that it must be her birthday. He asked her how old she was and she told him that yesterday her age was a square number and today it was a prime number. Would he be able to identify how old Suzie was?

Worked Solutions to Examples

South Korea Example

Using only solid or broken lines, what fraction of all the possible trigrams appear on the South Korean flag.

Solution

First we must ensure we understand the problem. Lines can be either solid or broken and their are three lines.

To be systematic, we could start with three solid lines and then gradually introduce broken lines.
If there were one broken line, there are three positions that it could be in.

If there were two broken lines, there are three positions that the remaining solid line could be in.

And if there were three broken lines, there is only one such arrangement. So all of the possible trigrams are:

As there are 8 in total and the flag contain 4, so the answer to the problem is that 1/2 of the possible trigrams appear on the flag.

Postman Example

Solution

We are interested in consecutive numbers where the first number is square and the second number is prime.

Because it is an age and (almost) all ages are from 1 to 121 we only need to check 11 square numbers to have checked all the square numbers that can be ages. For each we simply need to look at the following number to check if it is prime.

We look at the square numbers rather than the prime numbers, because there are fewer and it is easier to identify them. For instance square number 1 is followed by 2, which is prime, and square number 4 is followed by 5 which is prime. Continuing like this we can compile a table of our results:

From the table we can see that possible ages of Suzie are 2, 5, 17, 37 and 101. Hopefully he should be able to identify which of these specific ages was her actual one, based on her appearance.

18 questions in increasing order of difficulty