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.

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.

Substitution Cipher

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

Cola Can

Stage: 4 Challenge Level:

This printable worksheet may be useful: Cola Can.

Once the lower level thinking covered in the Hint has been assimilated students might be guided if necessary to see the value of a spreadsheet when solving a problem of this sort.

Additionally the use of a graph representing the spreadsheet values is particularly helpful for 'picturing' the behaviour of the surface area function as either base radius or can height varies.

There is a valuable opportunity to work with each of the two obvious independent variables : base radius and height. Starting with either of these the other is calculable from the specified volume of 330 ml, and once both r and h are known the surface area is calculable.