Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
The diagram shows an equilateral triangle dissected into 4 smaller equilateral triangles. . Which numbers of smaller triangles are possible and which are not, and why? This remains a Tough Nut as nobody has yet explained why it is impossible to dissect an equilateral triangle into certain numbers of smaller equilateral triangles.
Robert and Andrew, St James Middle School, Bury St Edmund's sent this solution with six equilateral triangles. They say they think this is the only solution with 6 smaller triangles other than rotating the shape.
Talei and Stephanie of Poltair Community School and Sports College, St Austell, Cornwall said they think that dissection into 2 or 3 smaller triangles is impossible and that 4 is the smallest number of triangles possible. They gave this as an example of dissection into 15 smaller triangles.
Talei's also had the idea of allowing overlapping triangles and realised that this would make it possible to have any number of smaller triangles, even infinitely many, but that would be a different question.