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?
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?
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.
Can you explain how this card trick works?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any 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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
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
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?
Delight your friends with this cunning trick! Can you explain how
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?
This challenge asks you to imagine a snake coiling on itself.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
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?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
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.
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.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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
An investigation that gives you the opportunity to make and justify
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
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?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?