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.
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.
Find the frequency distribution for ordinary English, and use it to help you crack the code.
For example, suppose the fire is half the distance from the river that the fire-fighter is. Try that the other way around as well.
Suppose the fire is still half the fire-fighter's distance but both are now nearer the river than before.
Or further away.
Suppose the fire is still half the fire-fighter's distance but the two positions are further from each other along the bank. Closer along the bank ?
Do one yourself with a calculator to help you see what calculation (formula) you need to make the spreadsheet do.