During the third hour after midnight the hands on a clock point in
the same direction (so one hand is over the top of the other). At
what time, to the nearest second, does this happen?
Alice's mum needs to go to each child's house just once and then
back home again. How many different routes are there? Use the
information to find out how long each road is on the route she
On a digital clock showing 24 hour time, over a whole day, how many
times does a 5 appear? Is it the same number for a 12 hour clock
over a whole day?
Stuart's watch loses two minutes every hour. Adam's watch gains one
minute every hour. Use the information to work out what time (the
real time) they arrived at the airport.
How could you arrange at least two dice in a stack so that the total of the visible spots is 18?
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
An investigation that gives you the opportunity to make and justify
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Can you find six numbers to go in the Daisy from which you can make
all the numbers from 1 to a number bigger than 25?
My cousin was 24 years old on Friday April 5th in 1974. On what day
of the week was she born?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Can you draw a square in which the perimeter is numerically equal
to the area?
What is the date in February 2002 where the 8 digits are
palindromic if the date is written in the British way?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
On a digital 24 hour clock, at certain times, all the digits are
consecutive. How many times like this are there between midnight
and 7 a.m.?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
This challenge extends the Plants investigation so now four or more children are involved.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
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.
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
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?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
Look carefully at the numbers. What do you notice? Can you make
another square using the numbers 1 to 16, that displays the same
Explore this how this program produces the sequences it does. What
are you controlling when you change the values of the variables?