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Euler found four whole numbers such that the sum of any two of the numbers 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

See the Hint section for detailed support for students, but the main aims of this problem are :

1. If a calculation approach is adopted, to see the value of using a spreadsheet . Thinking through the calculation required and the spreadsheet commands (formulae) necessary to achieve that are prime Stage 4 activities.

2. To account for the general result that emerges, in such a way that the 'solution' then becomes obvious, adds an additional visualisation to the student's repertoire of possibilities.