Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
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.
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 loser is the player who takes the last counter.
An account of some magic squares and their properties and and how to construct them for yourself.
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
An article which gives an account of some properties of magic squares.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
A collection of games on the NIM theme
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
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. . . .
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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. . . .
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.
Can you find the values at the vertices when you know the values on
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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 the area of a parallelogram defined by two vectors?
Find the vertices of a pentagon given the midpoints of its sides.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
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 diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
Can you find the values at the vertices when you know the values on
the edges of these multiplication arithmagons?
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
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 numbers from a sequence. What numbers can we make now?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?