Can you discover whether this is a fair game?
This is an interactivity in which you have to sort into the correct
order the steps in the proof of the formula for the sum of a
This is an interactivity in which you have to sort the steps in the
completion of the square into the correct order to prove the
formula for the solutions of quadratic equations.
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
Here the diagram says it all. Can you find the diagram?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?
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?
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Can you work through these direct proofs, using our interactive
Three equilateral triangles ABC, AYX and XZB are drawn with the
point X a moveable point on AB. The points P, Q and R are the
centres of the three triangles. What can you say about triangle
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
A point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
Some diagrammatic 'proofs' of algebraic identities and
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Prove Pythagoras Theorem using enlargements and scale factors.
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
Given that u>0 and v>0 find the smallest possible value of
1/u + 1/v given that u + v = 5 by different methods.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Peter Zimmerman from Mill Hill County High School in Barnet, London
gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is
divisible by 33 for every non negative integer n.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
Investigate the number of points with integer coordinates on
circles with centres at the origin for which the square of the
radius is a power of 5.
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
By proving these particular identities, prove the existence of general cases.
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
A introduction to how patterns can be deceiving, and what is and is not a proof.
Tom writes about expressing numbers as the sums of three squares.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Take a complicated fraction with the product of five quartics top
and bottom and reduce this to a whole number. This is a numerical
example involving some clever algebra.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Can you make sense of the three methods to work out the area of the kite in the square?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.