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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

First the fire :

It may help to draw your own diagram something like the one below. A fire-fighter needs to fill a bucket of water from the river and take it to the fire.

If, just as a first approximation, we don't worry about a filled bucket being heavier to carry, what is the best point on the river bank for the fire-fighter to fill the bucket ?.

If you need to do a calculation with lengths, what measurements will you need to make from your diagram ?

It's a general solution you are looking for, so you may need two or three different arrangements or diagrams to see how the solution relates to the positions of the fire-fighter and the fire.

Now the keys :

Draw a horizontal line. Fix two pins at different horizontal levels above the line. A set of keys slides on a string and the string runs over those two pins (the pins are not directly underneath one another). Gradually let out the string length until the weight of the keys brings the string down to touch the drawn horizontal line.

Can you predict where on that line the string will just touch, for different pin positions ?

Can you see the connection between this problem and the fire problem above ?