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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
This activity involves rounding four-digit numbers to the nearest thousand.
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?
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 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?
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.
Can you find the values at the vertices when you know the values on
Can you explain how this card trick works?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
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 work out how to win this game of Nim? Does it matter if you go first or second?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Delight your friends with this cunning trick! Can you explain how
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
Find out what a "fault-free" rectangle is and try to make some of
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
What happens when you round these three-digit numbers to the nearest 100?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
What happens when you round these numbers to the nearest whole number?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
Find the sum of all three-digit numbers each of whose digits is
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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 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. . . .
This challenge asks you to imagine a snake coiling on itself.
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from 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. . . .
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?
Here are two kinds of spirals for you to explore. What do you notice?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
It would be nice to have a strategy for disentangling any tangled
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
A collection of games on the NIM theme
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.
Can you describe this route to infinity? Where will the arrows take you next?