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?
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?
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
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
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.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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.
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.
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 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.
An investigation that gives you the opportunity to make and justify
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can you find the values at the vertices when you know the values on
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.
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 a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Charlie and Lynne put a counter on 42. They wondered if they could
visit all the other numbers on their 1-100 board, moving the
counter using just these two operations: x2 and -5. What do you
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?
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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?
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?
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 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?
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 moves does it take to swap over some red and blue frogs? Do you have a method?
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?