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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.

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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.

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Substitution Cipher

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

The Fire-fighter's Car Keys

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The fireman must stop at some point of the riverbank on his way to the fire. If we reflect the fire in the riverbank then our problem is equivalent to finding the shortest path, via the riverbank, to the reflected fire. This is just a straight line, so we reflect this line back in the riverbank to get our actual path. This shortest path is characterised by the fact that itarrives and leaves the riverbank at the same angle. We say that the angle of incidence and the angle of reflection are the same.

Can you solve the second, seemingly unrelated problem, using the first?