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In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?
How good are you at finding the formula for a number pattern ?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
Can you find the area of a parallelogram defined by two vectors?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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 a rule which connects consecutive triangular numbers?
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
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. . . .
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
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?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
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?
Show that all pentagonal numbers are one third of a triangular number.
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. . . .
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?
Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you make sense of these three proofs of Pythagoras' Theorem?
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?
How to build your own magic squares.
Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?
Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.