What happens when you try and fit the triomino pieces into these two grids?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that. . . .

I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only. . . .

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

Isabelle, Henry and Will describe some journeys that could fit the shape of the paths drawn.

Here are some imaginative stories that fit the graphs. Perhaps you can make up a different one?

Great reasoning from Alice who knows exactly when the speed is zero even if the motion doesn't stop in the way you might expect.

Jenny Murray describes the mathematical processes behind making patchwork in this article for students.

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.