How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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.
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.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
To avoid losing think of another very well known game where the patterns of play are similar.
A game for 2 players
Can you explain the strategy for winning this game with any target?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Delight your friends with this cunning trick! Can you explain how it works?
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?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
A game for 2 players with similarities 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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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 the values at the vertices when you know the values on the edges?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Got It game for an adult and child. How can you play so that you know you will always win?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
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?
Can all unit fractions be written as the sum of two unit fractions?
A collection of games on the NIM theme
Can you describe this route to infinity? Where will the arrows take you next?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
What's the largest volume of box you can make from a square of paper?
Can you find sets of sloping lines that enclose a square?
Can you use the diagram to prove the AM-GM inequality?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?