How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
An investigation that gives you the opportunity to make and justify
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Delight your friends with this cunning trick! Can you explain how
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
How many different journeys could you make if you were going to
visit four stations in this network? How about if there were five
stations? Can you predict the number of journeys for seven
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Find out what a "fault-free" rectangle is and try to make some of
Can you explain how this card trick works?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
What happens when you round these three-digit numbers to the nearest 100?
What happens when you round these numbers to the nearest whole number?
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
This activity involves rounding four-digit numbers to the nearest thousand.
Find the sum of all three-digit numbers each of whose digits is
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
One block is needed to make an up-and-down staircase, with one step
up and one step down. How many blocks would be needed to build an
up-and-down staircase with 5 steps up and 5 steps down?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
This challenge asks you to imagine a snake coiling on itself.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?