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.
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
What is the smallest number with exactly 14 divisors?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Can you find a way to identify times tables after they have been shifted up?
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
An investigation that gives you the opportunity to make and justify
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
Four of these clues are needed to find the chosen number on this
grid and four are true but do nothing to help in finding the
number. Can you sort out the clues and find the number?
Can you find any perfect numbers? Read this article to find out more...
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
How can you use just one weighing to find out which box contains
the lighter ten coins out of the ten boxes?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
A game that tests your understanding of remainders.
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
Substitution and Transposition all in one! How fiendish can these codes get?
Can you work out what size grid you need to read our secret message?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
Find the frequency distribution for ordinary English, and use it to help you crack the code.
Three people chose this as a favourite problem. It is the sort of
problem that needs thinking time - but once the connection is made
it gives access to many similar ideas.
Have a go at balancing this equation. Can you find different ways of doing it?
Given the products of adjacent cells, can you complete this Sudoku?
Can you work out some different ways to balance this equation?
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.