Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Label this plum tree graph to make it totally magic!
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
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.
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
How to build your own magic squares.
Find the five distinct digits N, R, I, C and H in the following
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 the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
Balance the bar with the three weight on the inside.
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?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
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. . . .
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
How good are you at finding the formula for a number pattern ?
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.
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. . . .
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?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
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?
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. . . .
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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. . . .
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. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
Can you find a rule which connects consecutive triangular numbers?
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?
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?