Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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 could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can you explain how this card trick works?
This challenge asks you to imagine a snake coiling on itself.
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
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?
This activity involves rounding four-digit numbers to the nearest thousand.
Delight your friends with this cunning trick! Can you explain how
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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?
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?
Find out what a "fault-free" rectangle is and try to make some of
Got It game for an adult and child. How can you play so that you know you will always win?
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. . . .
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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.
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
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.
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Here are two kinds of spirals for you to explore. What do you notice?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?