For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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.
This challenge asks you to imagine a snake coiling on itself.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Can you explain how this card trick works?
Delight your friends with this cunning trick! Can you explain how
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?
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?
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.
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.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
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?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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. . . .
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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. . . .
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. . . .
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.
This activity involves rounding four-digit numbers to the nearest thousand.
Explore the effect of reflecting in two intersecting mirror lines.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Got It game for an adult and child. How can you play so that you know you will always win?
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Can you find the values at the vertices when you know the values on
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?