Chapter 5
Notes for 22 August

Notes also available as PDF.

5.1 Review

5.1.1 Pólya’s problem solving principles

Principles are not a recipe, but they are a way to cook.

5.1.2 Tactic: Guessing

5.1.3 Tactic: Tabling / Making an orderly list

5.1.4 Example of creating a list

Problem 11 in set 1.2:

When Anita made a purchase she gave the clerk a dollar and received 21 cents in change. Complete this table to show what Anita’s change could have been.

# dimes# nickles# pennies
2 0 1

Removing extraneous details can help simplify a table-making method.

Everyone knows that debugging is twice as hard as writing a program in the first place. So if you’re as clever as you can be when you write it, how will you ever debug it? – Brian Kernighan (C, Unix, awk, …)

Ideal end results:

No number means a zero.

# dimes# nickles# penniestotal cents = 20
2 2*10 = 20
1 2 10+2*5 = 20
1 1 510+5+5*1 = 20
1 1010+10*1 = 20
4 4*5 = 20
3 53*5+5*1 = 20
2 102*5+10*1 = 20
1 151*5+15*1 = 20
2020*1 = 1

Remember: The “real” answer has one extra penny, so add 1 to the penny column.

5.1.5 Tactic not from the text: Orderly creation of a partial list

5.2 New tactic: Drawing a diagram

Talk about a coincidence, although clearly the example has no relationship to actual car models. Example 1.5 from the text, but done a little differently.

Example 1.5: Draw a diagram

In a stock car race the firist five finishers in some order were a Ford, a Pontiac, a Chevy, a Buick, and a Dodge.

  1. The Ford finished seven seconds before the Chevy.
  2. The Pontiac finished six seconds after the Buick.
  3. The Dodge finished eight seconds after the Buick.
  4. The Chevy finished two seconds before the Pontiac.

In what order did the cars finish the race?

5.2.1 Understanding the problem

What information do we have?

Is this enough information?

Try rephrasing the problem.

5.2.2 Devise a plan

5.2.3 Carry out the plan

Now we get to draw. Sorry, but I’m just using tables.

Using relationship 2 and 3,

D(8)-(7)-(6) -(5)-(4)-(3)-(2)-(1)B

Simplifying the presentation:


Place C by rule 4:


Now C is available, so place F by rule 1:


Or without the counts:


So the final finishing order is F, then B, then C, then P, and then D.

5.2.4 Looking back

5.3 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.