### Project 12.2

How can all the constructions of Euclidean be done by paper folding? What assumptions are made when paper is folded to construct geometric figures from Euclidean geometry?

What is orgami?

What is a hexaflexagon? How many flexes are possible?

What is a hexahexaflexagon? How many flexes are possible?

### Project 12.3

This problem illustrates a numerical solution for the Instant Insanity puzzle. Let’s associate numbers with the sides of the cubes of the Instant Insanity problem.

Let

White = 1;

Blue = 2;

Green = 3;

Red = 5

Now, the product across the top must be 30, and the product across the bottom must also be 30 (why?). It follows that a solution must have a product of 900 for the faces on the top and bottom. Consider the four cubes: Here are the products of top and bottom for the three possible arrangements of the first cube:

*Product*

Top: | Red = 5 | |

Bottom: | Red = 5 | 25 |

Top: | Blue = 2 | |

Bottom: | White = 1 | 2 |

Top: | Red = 5 | |

Bottom: | Green = 3 | 15 |

By a clever and systematic analysis of the products of top and bottom for cubes two, three, and four, you will find there are several ways of solving the top and bottom for a product of 900. Only one of these gives a product of 900 for front and back. This is the solution to the *Instant Insanity* puzzle. Find the solution using this method.

### Project 12.4

A puzzle sold under the name The Avenger, is pictured.

There are four problems posed in the article shown in the reference. Write a report on this article.

References:

“Group Theory, Rubik’s Cube and The Avenger,” *Games* , June/July 1987, pp. 44-45.

### Project 12.5

Consult one of the references and learn to solve Rubik’s cube.

Demonstrate your skill to the class. Nourse names the following categories:

20 minutes: WHIZ

10 minutes: SPEED DEMON

5 minutes: EXPERT

3 minutes: MASTER OF THE CUBE

- Stage a contest in front of the class to see which contestant can complete one face of a Rubik’s cube.
- Stage a contest to see who can solve the Rubik’s cube puzzle the fastest. Report the results to the class.

** References:**

**Ledbetter** and **Nering**,* The Solution to Rubik’s Cube *(Rohnert Park, CA, Noah’s Ark Enterprises, 1980).

**James G. Nourse**, *The Simple Solution to Rubik’s Cube *(New York: Bantam Books, 1981).

**David Singmaster**, *Notes on Rubik’s “Magic Cube,”* 5th ed. Hillside, N.J., Enslow Publishers, 1980).