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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
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.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
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 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.
Delight your friends with this cunning trick! Can you explain how
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
Can you find the values at the vertices when you know the values on
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?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
It starts quite simple but great opportunities for number discoveries and patterns!
This challenge asks you to imagine a snake coiling on itself.
Can you explain how this card trick works?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
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 ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
It would be nice to have a strategy for disentangling any tangled
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Find out what a "fault-free" rectangle is and try to make some of
Can all unit fractions be written as the sum of two unit fractions?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?