Explaining, convincing and proving

problem
Square corners

Age

7 to 11

Challenge level

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
problem
Shopping basket

Age

11 to 16

Challenge level

The items in the shopping basket add and multiply to give the same amount. What could their prices be?
problem
Reach 100

Age

7 to 11

Challenge level

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
problem
Thirty nine, seventy five

Age

11 to 14

Challenge level

We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can you discover what the five different types of coins are worth and how many of each we have saved?
problem
Disappearing square

Age

11 to 14

Challenge level

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. Do you have any interesting findings to report?
problem
More mathematical mysteries

Age

11 to 14

Challenge level

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try to explain why this works.
problem
The genie in the jar

Age

11 to 14

Challenge level

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?
problem
Is it magic or is it maths?

Age

11 to 14

Challenge level

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying them. Can you explain how they work?
problem
Rectangle tangle

Age

7 to 11

Challenge level

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?
problem
Cross-country race

Age

14 to 16

Challenge level

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?