Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
How many moves does it take to swap over some red and blue frogs? Do you have a method?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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?
Can you find the values at the vertices when you know the values on
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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!
Can you find sets of sloping lines that enclose a square?
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. . . .
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. . . .
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
It would be nice to have a strategy for disentangling any tangled
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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?
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?
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?
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
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.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
A collection of games on the NIM theme
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Can you tangle yourself up and reach any fraction?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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.