How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Find the sum of all three-digit numbers each of whose digits is
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Find out what a "fault-free" rectangle is and try to make some of
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
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"?
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
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.
An investigation that gives you the opportunity to make and justify
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?
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?
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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you find the values at the vertices when you know the values on
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
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?
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. . . .
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you find sets of sloping lines that enclose a square?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
In each of the pictures the invitation is for you to: Count what
you see. Identify how you think the pattern would continue.
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?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Imagine you have a large supply of 3kg and 8kg weights. How many of
each weight would you need for the average (mean) of the weights to
be 6kg? What other averages could you have?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit 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.
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.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
It would be nice to have a strategy for disentangling any tangled
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
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.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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.
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?