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.
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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 encourages you to explore dividing a three-digit number by a single-digit number.
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.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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
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.
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?
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 find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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?
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.
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. . . .
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?
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?
An investigation that gives you the opportunity to make and justify
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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. . . .
A collection of games on the NIM theme
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.
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.
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Can you find the values at the vertices when you know the values on
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?
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?
Charlie and Abi 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 think?