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?
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?
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.
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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 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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you explain the strategy for winning this game with any target?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Nim-7 game for an adult and child. Who will be the one to take the last counter?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Delight your friends with this cunning trick! Can you explain how
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
Here are two kinds of spirals for you to explore. What do you notice?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle 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
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?
Find out what a "fault-free" rectangle is and try to make some of
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Can you explain how this card trick works?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
This challenge asks you to imagine a snake coiling on itself.
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
An investigation that gives you the opportunity to make and justify
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, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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!
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.