Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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.
Delight your friends with this cunning trick! Can you explain how
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you tangle yourself up and reach any fraction?
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?
Can you find the values at the vertices when you know the values on
It would be nice to have a strategy for disentangling any tangled
Can you explain how this card trick works?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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. . . .
Charlie and Lynne 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
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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.
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Can you find the area of a parallelogram defined by two vectors?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
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?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can all unit fractions be written as the sum of two unit fractions?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
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?
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. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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. . . .
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?
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?
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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
Can you find sets of sloping lines that enclose a square?
It starts quite simple but great opportunities for number discoveries and patterns!
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?