Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
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?
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
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Find the sum of all three-digit numbers each of whose digits is
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?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
An investigation that gives you the opportunity to make and justify
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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
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?
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.
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Find out what a "fault-free" rectangle is and try to make some of
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
This challenge asks you to imagine a snake coiling on itself.
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?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
It starts quite simple but great opportunities for number discoveries and patterns!
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Delight your friends with this cunning trick! Can you explain how