Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Find the sum of all three-digit numbers each of whose digits is
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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.
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?
An investigation that gives you the opportunity to make and justify
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 out what a "fault-free" rectangle is and try to make some of
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do 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.
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
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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 work out how to win this game of Nim? Does it matter if you
go first or second?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
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?
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
In each of the pictures the invitation is for you to: Count what
you see. Identify how you think the pattern would continue.
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.
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?
Can you find sets of sloping lines that enclose a square?
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
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?
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. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
This challenge asks you to imagine a snake coiling on itself.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
A collection of games on the NIM theme
Can you find the values at the vertices when you know the values on
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?
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Can you explain how this card trick works?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Delight your friends with this cunning trick! Can you explain how
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?
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the