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 explain the strategy for winning this game with any target?
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Got It game for an adult and child. How can you play so that you know you will always win?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Delight your friends with this cunning trick! Can you explain how
Can you explain how this card trick works?
Find out what a "fault-free" rectangle is and try to make some of
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Here are two kinds of spirals for you to explore. What do you notice?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Can you find all the ways to get 15 at the top of this triangle of numbers?
This task follows on from Build it Up and takes the ideas into three dimensions!
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
This challenge asks you to imagine a snake coiling on itself.