Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Delight your friends with this cunning trick! Can you explain how
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
A game for 2 players
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?
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 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?
Can you explain how this card trick works?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
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
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?
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. . . .
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. . . .
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.
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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. . . .
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
It starts quite simple but great opportunities for number discoveries and patterns!
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find sets of sloping lines that enclose a square?
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?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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.
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
A collection of games on the NIM theme
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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?
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