Can you find the values at the vertices when you know the values on the edges?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
An article which gives an account of some properties of magic squares.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
Can you tangle yourself up and reach any fraction?
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
What's the largest volume of box you can make from a square of paper?
It would be nice to have a strategy for disentangling any tangled ropes...
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
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. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you use the diagram to prove the AM-GM inequality?
Make some loops out of regular hexagons. What rules can you discover?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
An account of some magic squares and their properties and and how to construct them for yourself.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
What is the total number of squares that can be made on a 5 by 5 geoboard?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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.
I added together some of my neighbours house numbers. Can you explain the patterns I noticed?
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.
A game for 2 players
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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 collection of games on the NIM theme
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
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?