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?
Delight your friends with this cunning trick! Can you explain how
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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 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?
Can you explain how this card trick works?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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
This challenge asks you to imagine a snake coiling on itself.
Find out what a "fault-free" rectangle is and try to make some of
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Here are two kinds of spirals for you to explore. What do you notice?
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
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
Find the sum of all three-digit numbers each of whose digits is
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?
What happens when you round these numbers to the nearest whole number?
This activity involves rounding four-digit numbers to the nearest thousand.
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
What happens when you round these three-digit numbers to the nearest 100?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you tangle yourself up and reach any fraction?
An investigation that gives you the opportunity to make and justify
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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 many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred 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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.