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. . . .
Can you explain how this card trick works?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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.
Delight your friends with this cunning trick! Can you explain how
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. . . .
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. . . .
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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. . . .
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
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 a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
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.
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. . . .
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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 see how to build a harmonic triangle? Can you work out the
next two rows?
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.
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.
Can you find the values at the vertices when you know the values on
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?
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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?
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?
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
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. . . .
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
Great Granddad is very proud of his telegram from the Queen
congratulating him on his hundredth birthday and he has friends who
are even older than he is... When was he born?
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. . . .
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.
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.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?