Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Can you cut up a square in the way shown and make the pieces into a triangle?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

Can you fit the tangram pieces into the outline of this plaque design?

A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

Investigate polygons with all the vertices on the lattice points of a grid. For each polygon, work out the area A, the number B of points on the boundary and the number of points (I) inside. . . .

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Basic strategy games are particularly suitable as starting points for investigations. Players instinctively try to discover a winning strategy, and usually the best way to do this is to analyse the outcomes of series of 'moves'. With a little encouragement from the teacher, a mathematical investigation is born.