### Project 8.2 **A 51-Star Flag**

Recently there has been talk of Puerto Rico becoming the 51st state. If that happens, we will need to have a flag with 51 stars. Design a flag with 51 stars discussing the pros and cons of various possibilities.

### Project 8.3

Write a paper on Pythagorean triples (numbers *a*, *b*, and *c* that satisfy the Pythagorean theorem). For example, pick an odd positive integer; say 9.

Square it: 81.

Find consecutive integers that have the sum 81: 40 and 41.

You have generated a Pythagorean triple: 9, 40, 41

9

^{2}+ 40^{2}= 41^{2}### Project 8.4

Look at Project 8.3, and extend the idea of a Pythagorean triplet. A Pythagorean quartet are four numbers *a*, *b*, *c* and *d* so that

*a*

^{2}+

*b*

^{2}+

*c*

^{2}=

*d*

^{2}

Outline a method for finding Pythagorean quartets.

What do you think is meant by a Pythagorean pentad?