You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?
A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?
How many different triangles can you make on a circular pegboard that has nine pegs?
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
What is the best way to shunt these carriages so that each train can continue its journey?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?
One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
Can you fit the tangram pieces into the outline of Granma T?
Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?
Can you fit the tangram pieces into the outline of the child walking home from school?
Can you fit the tangram pieces into the outlines of these clocks?
Can you fit the tangram pieces into the outline of the telescope and microscope?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
Can you fit the tangram pieces into the outlines of the chairs?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Can you cut up a square in the way shown and make the pieces into a triangle?
Here's a simple way to make a Tangram without any measuring or ruling lines.
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Can you fit the tangram pieces into the outline of Mai Ling?
Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?
Can you fit the tangram pieces into the outline of these rabbits?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Can you fit the tangram pieces into the outline of Little Fung at the table?
Have a go at this 3D extension to the Pebbles problem.
This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .