Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
What happens when you round these numbers to the nearest whole number?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Find the sum of all three-digit numbers each of whose digits is
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
What happens when you round these three-digit numbers to the nearest 100?
Delight your friends with this cunning trick! Can you explain how
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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?
Can you explain how this card trick works?
This activity involves rounding four-digit numbers to the nearest thousand.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Ben’s class were making cutting up number tracks. First they
cut them into twos and added up the numbers on each piece. What
patterns could they see?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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. . . .
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
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
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?
An investigation that gives you the opportunity to make and justify
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
A collection of games on the NIM theme