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?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
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?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Here is a chance to play a version of the classic Countdown Game.
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....?
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?
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?
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 find all the different ways of lining up these Cuisenaire rods?
Find out what a "fault-free" rectangle is and try to make some of your own.
If you have only four weights, where could you place them in order to balance this equaliser?
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?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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!
Try out the lottery that is played in a far-away land. What is the chance of winning?
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
Can you find all the different triangles on these peg boards, and find their angles?
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?
An environment which simulates working with Cuisenaire rods.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.
Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
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?
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?
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.
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?
Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?
How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?
Choose a symbol to put into the number sentence.
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?
Can you explain the strategy for winning this game with any target?
Use the Cuisenaire rods environment to investigate ratio. Can you find pairs of rods in the ratio 3:2? How about 9:6?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Can you use the interactive to complete the tangrams in the shape of butterflies?