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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.

Peaches in General

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

Why do this problem?

This an excellent problem through which to give students an opportunity to experience iterative processes. It also moves, by easy stages, into a multi-variable problem, and the value of using a spreadsheet is readily apparent.

Possible approach

Starting with the initial given problem discuss solutions found by the group.

Invite ideas about possible directions for generalisation, perhaps starting with the easier results like allowing 'plus one more' to become plus two, plus three, and so on.

Clarify what 'result' has actually been discovered for each generalisation and spend plenty of time letting students sense the 'mathematical need' to account for each 'result'. These are good questions to be 'left in the air', allowing students to turn these over in their minds over time.

Key questions

  • What generalisations are possible ?
  • If we explore those one at a time, which shall we take first ?
  • What general statement can we now make ?
  • Can we justify that ? Explaining why it should be so.

Possible extension

More able students will produce more extended generalisations and have a motivation to account for what is observed, challenging one another to communicate clear explanations or visualisations of the fundamental processes.

Able students will sense the potential power of a spreadsheet and should be encouraged to work collaboratively to become proficient and confident in its use.

Possible support

The Stage 3 problem is sufficient as it is, to offer less able students a rich experience producing results that form a clear pattern which will not have been obvious at the start.