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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.

Funnel

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

See the Hint section for detailed support for students, but one of the main aims of this problem is to use a spreadsheet to open up a problem that is otherwise rather tedious to pursue by calculation. Students may need help thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that. This is a challenging problem.

The relationship between surface area and volume enclosed is an important theme for able students at Stage 4 and using IT effectively should be part of a good student's repertoire of problem solving approaches.

With able students it is good to draw out that the ratio between the volume and the surface area will change with scale, but the optimum shape for the cone will not.
Perhaps reasoning in terms of units : measurements in a different unit of length would change all the numbers (how ?) but not the height-radius ratio (or cone angle) where the surface area takes its minimum value.