How many lines are there in your diagrams? How many lines are there in the original diagram? Can you find a way of working out […]
Five red-eye frogs and five orange frogs line up in a row with a space between them.
They have to change places but they can only hop, one frog over another frog or slide to an empty space next to them.
Red-eye frogs can […]
This is a game for two players (or a class split into two teams).
Take it in turns to choose one of the numbers 1 to 9. It is now your number and your opponent cannot choose it. Each number can be chosen only once.
To win, be the first to pick 3 numbers that […]
between the two 1s there is one digit giving 1?1,
between the two 2s there are two digits giving 2??2,
between the two 3s there are three digits giving 3???3.
It is not difficult to put a list of decimals in order of size. But what about ordering fractions?
John Farey introduced sequences of sequences of fractions in order of size, now called Farey sequences. Can you discover his method from the following examples?
What patterns do you see in them?
Suppose you have to share 9872 gold coins between 8 people.
You give the coins out in thousands until there are less than 8000 left.
Then you give out one hundred at a time until there are less than 800, then […]
He labelled each bag with the number of coins inside it.
He could then pay any sum of money from 1 coin to 15 coins using one or more of the bags without opening any of the bags.
Give a proof linked to this picture of the formula $a^2 – b^2 = (a + b)(a – b)$ .
2. What do you notice about the difference between squares of consecutive numbers?
Compare the differences between squares of numbers that differ by 1 […]
This picture shows the first 4 triangle numbers: 1, 3, 6 and 10 and how you can fit 2 identical copies of a triangle together to form a rectangle.
Can you use it to find the number of dots in the […]
Is there more than one?
It is quite easy to find a few solutions. The big challenge is to find all possible solutions.
Can you provide a convincing argument that you have found […]