Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!
If you have only four weights, where could you place them in order to balance this equaliser?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Choose a symbol to put into the number sentence.
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?
What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?
Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?
Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?
Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?
Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?
Here is a chance to play a version of the classic Countdown Game.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
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?
Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?
This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?
Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?
A generic circular pegboard resource.
A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!
How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
Use the interactivities to complete these Venn diagrams.
A simulation of target archery practice
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
What shaped overlaps can you make with two circles which are the same size? What shapes are 'left over'? What shapes can you make when the circles are different sizes?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
Can you find all the different ways of lining up these Cuisenaire rods?
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Can you complete this jigsaw of the multiplication square?
Use the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards?
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
Use the interactivity or play this dice game yourself. How could you make it fair?
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?
What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.
Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.