### Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

### Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

### Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

##### Stage: 4 Challenge Level:
This problem developed from a question posed on the NRICH Projects site.

The mapping below was created using the NRICH Number Plumber.

Click here to watch a video explaining how to use the Number Plumber.

Click on the mapping to explore the relationship between the function and its graph.

Now replace the 1 and 3 with other values. How does this affect the graph?