Take three differently coloured blocks - maybe red, yellow and
blue. Make a tower using one of each colour. How many different
towers can you make?
Noah saw 12 legs walk by into the Ark. How many creatures did he see?
These alphabet bricks are painted in a special way. A is on one
brick, B on two bricks, and so on. How many bricks will be painted
by the time they have got to other letters of the alphabet?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Where can you draw a line on a clock face so that the numbers on
both sides have the same total?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
A church hymn book contains 700 hymns. The numbers of the hymns are
displayed by combining special small single-digit boards. What is
the minimum number of small boards that is needed?
Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
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?
Which set of numbers that add to 10 have the largest product?
What is the smallest number with exactly 14 divisors?
You are only given the three midpoints of the sides of a triangle.
How can you construct the original triangle?
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Investigate the sequences obtained by starting with any positive 2
digit number (10a+b) and repeatedly using the rule 10a+b maps to
10b-a to get the next number in the sequence.
Some relationships are transitive, such as `if A>B and B>C
then it follows that A>C', but some are not. In a voting system,
if A beats B and B beats C should we expect A to beat C?
Two tangents are drawn to the other circle from the centres of a
pair of circles. What can you say about the chords cut off by these
tangents. Be patient - this problem may be slow to load.
Can you make sense of these logical contortions?
Is it true that a large integer m can be taken such that: 1 + 1/2 +
1/3 + ... +1/m > 100 ?
Christina and Joanna worked out what the shapes had in common.
Christina has also suggested some sets of her own.
We received many detailed solutions to this challenge, most of
which made use of a spreadsheet.
It would be great to receive some more conjectures, and some data
which back up these conjectures.
Nice explanations from Francesca and from Stephen
What are rich tasks and contexts and why do they matter?
The very problem with problems, namely that they should result in
you being stuck, is at the heart of what problem-solving is about.
In this article for teachers I talk about just a few of the other
problems with problems that make them such a rich source of
The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas can be combined with new discoveries. prove a mathematical theorem.
This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.
Take a look at the steps involved in thinking through a problem.