### Have You Got It?

Can you explain the strategy for winning this game with any target?

### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### Card Trick 2

Can you explain how this card trick works?

# Pick's Theorem

##### Stage: 3 Challenge Level:

When the dots on square dotty paper are joined by straight lines the enclosed figures have dots on their perimeter ($p$) and often internal ($i$) ones as well.

Figures can be described in this way: $(p, i)$.
For example, the red square has a $(p,i)$ of $(4,0)$, the grey triangle $(3,1)$, the green triangle $(5,0)$ and the blue hexagon $(6,4)$:

Each figure you produce will always enclose an area ($A$) of the square dotty paper.

The examples in the diagram have areas of $1$, $1 {1 \over 2}$, and $6$ sq units.

Check that you agree.

Draw more figures and keep a record of their perimeter points ($p$), interior points ($i$) and areas ($A$).

Can you find a relationship between these three variables?