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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Regular Hexagon Loops

Age 11 to 14 Challenge Level:

A hexagon loop is a closed chain of hexagons that meet along a whole edge and in which each hexagon must touch exactly two others.


They do not need to be symmetrical or short:

Can you find any rules connecting the numbers of tiles, the inside perimeter and the outside perimeter?

You might want to start by exploring square loops or growing sequences like these:

If you haven't got a supply of regular hexagons you could work on these regular hexagon sheets

With thanks to Don Steward, whose ideas formed the basis of this problem.