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?

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.

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

Age 14 to 16 Challenge Level:

This problem is available as a printable worksheet: Beelines.

Why do this problem?

Working on this problem will give students an opportunity to make and justify conjectures that link numerical and geometrical ideas. It offers students valuable experience of working on coordinate geometry and will develop their understanding of factors and multiples.

Show the video below, or introduce the problem in the same way. You could use this Geogebra applet or download this Geogebra file.

"In a while I'm going to give you the coordinates of the point at the end of a line, and challenge you to tell me how many squares the line will pass through."

Students could work in pairs, choosing coordinates and keeping a record of their results.
Once students have a selection of results, invite them to look for patterns.

If they find it hard to spot any relationships, here are some prompts:
"Group together all your results where the x and y coordinate are the same. What do you notice? Will it always happen?"
"What about results where the x coordinate is twice the y coordinate? Or three times?"
"What about results when the two coordinates have no factors in common?"

To finish off, choose some pairs of coordinates and challenge students to work out quickly how many squares the line will pass through. This is a very good activity for students to display their answers on mini-whiteboards.

Finally, take some time to discuss the geometrical reasons why a straight line joining the origin to a point with coprime coordinates will not go through any grid squares, and the reason why we are able to calculate the number of squares the lines will go through without needing to draw and count.

Possible extension

This Wikipedia article about line-drawing algorithms may be of interest to students who are keen on computer programming.
You could challenge students to find a way to work out whether a given line passes through a given square on the grid (without drawing!).