Can you explain how this card trick works?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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. . . .
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you find the values at the vertices when you know the values on
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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 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?
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?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
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. . . .
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Explore the effect of reflecting in two intersecting mirror lines.
Can you find sets of sloping lines that enclose a square?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Explore the effect of combining enlargements.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can all unit fractions be written as the sum of two unit fractions?
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
Explore the effect of reflecting in two parallel mirror lines.
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?