### Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

### Euclid's Algorithm I

How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with the solutions x and y being integers? Read this article to find out.

### What's a Group?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

# Eyes Down

### Why do this problem?

The problem gives practice in working with linear inequalities and in working systematically through separate cases.

### Possible approach

Encourage the class to try some numerical values for $x$, to compare the values of the three functions and to record their findings. Collect sufficient results from the class to provide evidence for spotting patterns and making conjectures.

### Key questions

If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a whole number and $0\leq b < 1$. What is the difference between the separate cases where $0 \leq b < {1\over 2}$ and ${1\over 2}\leq b < 1$?