Have a go at this 3D extension to the Pebbles problem.
What are the next three numbers in this sequence? Can you explain
why are they called pyramid numbers?
I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Your challenge is to find the longest way through the network
following this rule. You can start and finish anywhere, and with
any shape, as long as you follow the correct order.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Explore the different tunes you can make with these five gourds.
What are the similarities and differences between the two tunes you
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
How do you know if your set of dominoes is complete?
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
How many different ways can I lay 10 paving slabs, each 2 foot by 1
foot, to make a path 2 foot wide and 10 foot long from my back door
into my garden, without cutting any of the paving slabs?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
An environment which simulates working with Cuisenaire rods.
Make some intricate patterns in LOGO
Find the next number in this pattern: 3, 7, 19, 55 ...
Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
Investigate what happens when you add house numbers along a street
in different ways.
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
Make some loops out of regular hexagons. What rules can you discover?
If the numbers 5, 7 and 4 go into this function machine, what
numbers will come out?
What is the total area of the first two triangles as a fraction of
the original A4 rectangle? What is the total area of the first
three triangles as a fraction of the original A4 rectangle? If. . . .
"Tell me the next two numbers in each of these seven minor spells",
chanted the Mathemagician, "And the great spell will crumble away!"
Can you help Anna and David break the spell?
Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.
What's the greatest number of sides a polygon on a dotty grid could have?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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
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?
Choose any 4 whole numbers and take the difference between
consecutive numbers, ending with the difference between the first
and the last numbers. What happens when you repeat this process
over and. . . .
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When
did July 1st fall on a Monday again?
In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?
Explore one of these five pictures.
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Investigate the successive areas of light blue in these diagrams.
In this investigation, you are challenged to make mobile phone
numbers which are easy to remember. What happens if you make a
sequence adding 2 each time?
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?
There are lots of ideas to explore in these sequences of ordered
Explore this how this program produces the sequences it does. What
are you controlling when you change the values of the variables?
Three people chose this as a favourite problem. It is the sort of
problem that needs thinking time - but once the connection is made
it gives access to many similar ideas.
Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
How many different sets of numbers with at least four members can
you find in the numbers in this box?
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?