Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
To avoid losing think of another very well known game where the patterns of play are similar.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Find the vertices of a pentagon given the midpoints of its sides.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
A game for 2 players
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
A collection of games on the NIM theme
Can you find the area of a parallelogram defined by two vectors?
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
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.
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
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?
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Can you describe this route to infinity? Where will the arrows take you next?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?