Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

How would you count the number of fingers in these pictures?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

Susie took cherries out of a bowl by following a certain pattern. How many cherries had there been in the bowl to start with if she was left with 14 single ones?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Dotty Six is a simple dice game that you can adapt in many ways.

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Investigate what happens when you add house numbers along a street in different ways.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

Dotty Six game for an adult and child. Will you be the first to have three sixes in a straight line?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Jack's mum bought some candles to use on his birthday cakes and when his sister was born, she used them on her cakes too. Can you use the information to find out when Kate was born?

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

There were 22 legs creeping across the web. How many flies? How many spiders?

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

Woof is a big dog. Yap is a little dog. Emma has 16 dog biscuits to give to the two dogs. She gave Woof 4 more biscuits than Yap. How many biscuits did each dog get?

Twizzle, a female giraffe, needs transporting to another zoo. Which route will give the fastest journey?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.