Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Can you describe this route to infinity? Where will the arrows take you next?

Collect as many diamonds as you can by drawing three straight lines.

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Weekly Problem 45 - 2011

What shapes can be made by folding an A4 sheet of paper only once?

Weekly Problem 52 - 2011

Draw two intersecting rectangles on a sheet of paper. How many regions are enclosed? Can you find the largest number of regions possible?

Weekly Problem 3 - 2012

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

Weekly Problem 17 - 2012

Which shapes can be made by folding a piece of A4 paper?

Weekly Problem 52 - 2012

An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

Weekly Problem 34 - 2013

A card with the letter N on it is rotated through two different axes. What does the card look like at the end?

Weekly Problem 20 - 2006

A single piece of string is threaded through five holes on a piece of card. How is this possible?

Weekly Problem 28 - 2007

A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?

Weekly Problem 43 - 2007

The diagram shows 10 identical coins which fit exactly inside a wooden frame. What is the largest number of coins that may be removed so that each remaining coin is still unable to slide.

Weekly Problem 46 - 2007

When a solid cube is held up to the light, how many of the shapes shown could its shadow have?

Weekly Problem 25 - 2008

An ordinary die is placed on a horizontal table with the '1' face facing East... In which direction is the '1' face facing after this sequence of moves?

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

Weekly Problem 19 - 2009

When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?

Weekly Problem 34 - 2009

I am standing behind five pupils who are signalling a five-digit number to someone on the opposite side of the playground. What number is actually being signalled?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Weekly Problem 47 - 2009

How many squares can you draw on this lattice?

Weekly Problem 18 - 2010

Three faces of a $3 \times 3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?

Weekly Problem 37 - 2010

An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?

Weekly Problem 24 - 2011

Can you find the time between 3 o'clock and 10 o'clock when my digital clock looks the same from both the front and back?

Weekly Problem 29 - 2011

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Weekly Problem 4 - 2014

What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?

Weekly Problem 52 - 2014

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

Weekly Problem 12 - 2015

Eight lines are drawn in a regular octagon to form a pattern. What fraction of the octagon is shaded?

Weekly Problem 18 - 2015

Beatrix relfects the letter P in all three sides of a triangle in turn. What is the final result?

Weekly Problem 34 - 2015

Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

Weekly Problem 9 - 2016

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

Weekly Problem 24 - 2016

What is the smallest number of additional lines that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?

Weekly Problem 43 - 2016

In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

Weekly Problem 27 - 2017

A cube is rolled on a plane, landing on the squares in the order shown. Which two positions had the same face of the cube touching the surface?

Weekly Problem 48 - 2017

What is the surface area of the solid shown?