Chapter 4
Notes for 20 August

Notes also available as PDF.

4.1 Review

4.1.1 Problem solving

4.1.2 Categories

4.2 Today’s goal: Problem solving principles

4.3 Pólya’s principles

These are principles and not a recipe. Use these to form a problem-solving plan. (problem solving as a problem…)

4.3.1 Understand the problem

4.3.2 Divise a plan

Sure I’m lucky. And the more I practice, the luckier I get. – Gary Player, golfer

4.3.3 Carry out the plan

4.3.4 Examine your solution

4.4 Two closely related tactics, guessing and making a list

4.4.1 Guessing

4.4.2 Example 1.3 from the text

4.4.3 Tabling / Making a list

4.4.4 Example 1.4 from the text: a counting problem.

When you read it, note the follow,

Example 1.4, page 15 (?):

Make an orderly list

How many different total scores could you make if you hit the dartboard shown with three darts?

(Three nested circles. Scores 10 for the innermost, 5 for the middle, and 1 for the outer ring.)

Understand the problem

Three darts hit the dartboard and each scores a 1, 5, or 10. The total score is the sum of the scores for the three darts. There could be three 1s, two 1s and a 5, one 5 and two 10s, and so on. The fact that we are told to find the total score when throwing three darts is just a way of asking what sums can be made using three numbers, each of which is either 1, 5, or 10.

Devise a plan

If we just write down sums hit or miss, we will almost surely overlook some of the possibilities. Using an orderly scheme instead, we can make sure that we obtain all possible scores. Let’s make such a list. We first list the score if we have three 1s. then two 1s and one 5, then two 1s and no 5s, and so on. In this way, we can be sure that no score is missed.

Carry out the plan

Number of 1sNumber of 5sNumber of 10sTotal Score
3 0 0 3
2 1 0 7
2 0 1 12
1 2 0 11
1 1 1 16
1 0 2 21
0 3 0 15
0 2 1 20
0 1 2 25
0 0 3 30

The possible total scores are listed.

Look back

Here the key to the solution was in being very systematic. We were careful first to obtain all possible scored with three 1s, then two 1s, then no 1s. With two 1s there could be either a 5 or a 10 as shown. For one 1 the only possibilities are two 5s and no 10s, one 5 and one 10, or no 5s and two 10s. Constructing the table in this orderly way makes it clear that we have not missed any possibilities.

My additional notes:

Note the similarity with numbers. Consider listing all numbers that fit the following two properties:

  1. Use only the digits 0, 1, 2, and 3.
  2. Have at most three digits.

Then drop those numbers whose digits do not add to three.

That’s another way to construct a table like this: Consider a larger table where it is easier to be systematic, then remove numbers that do not fit the problem.

4.4.5 Different example to show bisection

Modified problem 12.a from the text’s chapter review exercises.

Geometric progression
Sequence of numbers defined by a starting number and a constant. The second number is generated by multiplying by the constant, the third by multiplying again.

Consider the sequence where 3 is the starting number and two is the constant.

Which term in the sequence is 768?

4.5 Next time: More problem solving ideas.

4.6 Homework

Groups are fine, turn in your own work. Homework is due in or before class on Mondays.

Write out (briefly) your approach to each problem.

Note that you may email homework. However, I don’t use MicrosoftTM products (e.g. Word), and software packages are notoriously finicky about translating mathematics.

If you’re typing it (which I advise just for practice in whatever tools you use), you likely want to turn in a printout. If you do want to email your submission, please produce a PDF or PostScript document.