### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

### Nim

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

### Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

# Beelines

##### Stage: 4 Challenge Level:

Take a look at the video below:

If you can't see the video, click below for a description.

If I choose the point (5, 5) and draw a line segment joining the point to the origin, my line passes through 5 grid squares.
If I choose the point (4, 3) and draw a line segment joining the point to the origin, my line passes through 6 grid squares.
If I choose the point (6, 4) and draw a line segment joining the point to the origin, my line passes through 8 grid squares.

Draw some line segments of your own, and record how many grid squares each one passes through.

You may wish to explore this using this GeoGebra applet.

Can you find a relationship between the coordinates of the end of the line segment and the number of squares it passes through?

If I draw the line segment joining the origin to the point (50, 37) how many grid squares will it pass through?

If I draw the line segment joining the origin to the point (96, 72) how many grid squares will it pass through?

Can you find a line segment that passes through exactly 24 squares?
Can you find more than one?

Can you work out how many grid squares a line segment passes through, if you are given the coordinates of the two endpoints, where neither is at the origin?

You could also investigate the number of grid lines crossed...

Notes and Background

Working out which grid squares a straight line crosses allows you to create algorithms for drawing straight lines on a computer, where each pixel is a grid square. Read more about line drawing algorithms here.