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 NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Got It game for an adult and child. How can you play so that you know you will always win?
Can you explain the strategy for winning this game with any target?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
This activity involves rounding four-digit numbers to the nearest thousand.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Delight your friends with this cunning trick! Can you explain how
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you explain how this card trick works?
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?
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle 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.
Here are two kinds of spirals for you to explore. What do you notice?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
This challenge asks you to imagine a snake coiling on itself.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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.
Can you find all the ways to get 15 at the top of this triangle of 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.
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?
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.
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. . . .
Are these statements always true, sometimes true or never true?
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
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Find the sum of all three-digit numbers each of whose digits is
Find out what a "fault-free" rectangle is and try to make some of
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.
This task follows on from Build it Up and takes the ideas into three dimensions!