How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
This activity involves rounding four-digit numbers to the nearest thousand.
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
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. . . .
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
Got It game for an adult and child. How can you play so that you know you will always win?
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. . . .
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.
An investigation that gives you the opportunity to make and justify
Delight your friends with this cunning trick! Can you explain how
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you explain how this card trick works?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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
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?
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?
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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find the sum of all three-digit numbers each of whose digits is
What happens when you round these three-digit numbers to the nearest 100?
This challenge asks you to imagine a snake coiling on itself.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind