You may also like

problem icon

Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square. Three of the numbers that he found are a = 18530, b=65570, c=45986. Find the fourth number, x. You could do this by trial and error, and a spreadsheet would be a good tool for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then focus on Q^2-R^2=b-c which is known. Moreover you know that Q > sqrtb and R > sqrtc . Use this to show that Q-R is less than or equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and x for values of Q-R from 1 to 41 , and hence to find the value of x for which a+x is a perfect square.

problem icon

Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

problem icon

Substitution Cipher

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Cola Can

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A millilitre is one cubic centimetre.

Check that you can calculate the volume of a cylinder.

If the diameter is 6 cm, you can calculate the base area of the can.

If you know the base area you can find a height which will give you a specific volume ( 330 ml in this case)

Now start with a height (10 cm) and work your way back to a base area for a specified volume, then find a radius for that base area, and hence a diameter for the can.

What does least aluminium require ?

The top and base of the can are circles but how do you calculate the curved surface area ?