Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

This article for primary teachers suggests ways in which to help children become better at working systematically.

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

How many trapeziums, of various sizes, are hidden in this picture?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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. . . .

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

If you had 36 cubes, what different cuboids could you make?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

These practical challenges are all about making a 'tray' and covering it with paper.

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

How many possible necklaces can you find? And how do you know you've found them all?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

Use the numbers and symbols to make this number sentence correct. How many different ways can you find?