Chapter 8
Notes for 27 August

Notes also available as PDF.

8.1 Review: Pólya’s problem-solving principles

These are principles and not a recipe or a plan. Use these to form a problem-solving plan. (Problem solving itself is a problem…).

Previous tactics:

Today, a few more tactics:

8.2 Effective trial and error by bisection


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

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

1 3 = 3 ⋅ {2}^{0} = 3 ⋅ {2}^{n-1}
2 6 = 3 ⋅ {2}^{1}
3 12 = 3 ⋅ {2}^{2}

Which term in the sequence is 768?

8.2.1 Understanding the problem

8.2.2 Forming plans

8.2.3 Carrying out the new plan

8.2.4 Looking back

8.3 Simpler sub-problems for finding patterns

What is the units digit of {7}^{100}?

Number Expanded Last digit
{7}^{0} 1 1
{7}^{1} 7 7
{7}^{2} 49 9
{7}^{3} 343 3
{7}^{4} 2401 1
{7}^{5} 7
{7}^{6} 9

Looking back:

8.4 Other sources for tactics and examples

8.5 Next time: Reading graphs and charts

Recommended reading: Anything by Edward Tufte. The “Ask E.T.” section of his personal site ( has examples of excellent and poor graphics.

8.6 Homework

Practice is absolutely critical in this class.

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

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.