Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
What is the largest number of circles we can fit into the frame
without them overlapping? How do you know? What will happen if you
try the other shapes?
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
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.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
If the answer's 2010, what could the question be?
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
What happens to the area of a square if you double the length of
the sides? Try the same thing with rectangles, diamonds and other
shapes. How do the four smaller ones fit into the larger one?
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?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
What do these two triangles have in common? How are they related?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
An investigation that gives you the opportunity to make and justify
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
These pictures show squares split into halves. Can you find other ways?
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?
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
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
Investigate the number of faces you can see when you arrange three cubes in different ways.
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
You cannot choose a selection of ice cream flavours that includes
totally what someone has already chosen. Have a go and find all the
different ways in which seven children can have ice cream.
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
The red ring is inside the blue ring in this picture. Can you
rearrange the rings in different ways? Perhaps you can overlap them
or put one outside another?
In how many ways can you stack these rods, following the rules?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
How many tiles do we need to tile these patios?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
What is the largest cuboid you can wrap in an A3 sheet of paper?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Sort the houses in my street into different groups. Can you do it in any other ways?
I like to walk along the cracks of the paving stones, but not the
outside edge of the path itself. How many different routes can you
find for me to take?
Investigate what happens when you add house numbers along a street
in different ways.
An activity making various patterns with 2 x 1 rectangular tiles.
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
Explore ways of colouring this set of triangles. Can you make
This challenge encourages you to explore dividing a three-digit number by a single-digit number.