### Converse

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

### 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.

# Alison's Mapping

##### Age 14 to 16 Challenge Level:

Before starting to explore Alison's mapping, you might want to watch a short video explaining how to use the NRICH Number Plumber.

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Alison has created a mapping which you can access by clicking on the picture below. The initial challenge is to figure out what Alison's mappings do. You can drop some numbers into the mappings, and see what comes out.

Test your ideas by predicting the outputs for some different inputs.
Only reveal the hidden parts of the mapping once you are certain you know what is going on.
Can you find an input which gives the same output for both mappings?

Click on the picture to explore Alison's mappings.

As you explore Alison's mappings, you will notice points appearing on the graph. The input number is the x coordinate, and the output number is the y coordinate.
Create some other function machines using the NRICH Number Plumber, and observe the outcomes on the graph.