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The sums of the squares of three related numbers is also a perfect square - can you explain why?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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. . . .
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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?
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. . . .
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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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. . . .
You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?
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.
Which set of numbers that add to 10 have the largest product?
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?
Can you make sense of these three proofs of Pythagoras' Theorem?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
A huge wheel is rolling past your window. What do you see?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
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?
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. . . .
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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?
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
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.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?