Problem: Password strength

Passwords are something about which almost everyone needs to be better informed. As part of a unit on combinatorics (or alternatively, as a unit on passwords in a tech class), students could look at passwords and how to make passwords more secure.

To get students thinking about password strength, this interactive password haystack calculator would be useful. Students could start by trying to make some secure passwords through the interactive calculator, and then they would probably have questions (like: Why is this password so much more secure than this other password?).

This list of the 25 most commonly used passwords is also useful to start some conversation on the difference between password haystack and password strength.

Open ended problem: Dots and lines

In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.

This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?