Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Can all unit fractions be written as the sum of two unit fractions?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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. . . .
This challenge asks you to imagine a snake coiling on itself.
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
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?
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
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's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you describe this route to infinity? Where will the arrows take you next?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find out what a "fault-free" rectangle is and try to make some of
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 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.
A collection of games on the NIM theme
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
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?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
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?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Delight your friends with this cunning trick! Can you explain how
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?