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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.

Under the Ribbon

Stage: 4 Challenge Level: Challenge Level:1

One reason for using this problem with students is to show how a spreadsheet approach can give access to problems where other analysis, in this case algebra, is very difficult.

A second reason is that the answer to the problem is surprising.

Somehow the brain doesn't expect such a large answer - perhaps we imagine that because there isn't much extra length extending the ribbon ( 4 cm of slack ) there will not be much scope for displacing the ribbon vertically.

This problem also needs approaching in stages, and it is good for students to experience that by solving for the symmetric central position first then giving consideration to the asymmetric case.

Perhaps students could extend the enquiry to consider how different degrees of slack : 3 cm, 2 cm, 1 cm, effect the result.

Some students will be very familiar with using a spreadsheet to help with this kind of mathematical context, but many will not be at all confident, or even think to try a spreadsheet.

Making calculations for several different cube sizes on a calculator first, helps students to see that the calculation process is the same each time only using different input values. It also helps students to grasp that the ability to cover input values at intervals systematically across a chosen range could guide us closer and closer towards an optimal position or value.