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.

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.

Substitution Cipher

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Dating Made Easier

Stage: 4 Challenge Level:

If a sum invested gains $10\%$ each year how long will it be before it has doubled its value?

If an object depreciates in value by $10\%$ each year how long will it take until only half of the original value remains?

Why aren't these two answers the same?

Is there a rate, used for both gain and depreciation, for which those two answers would actually be the same?

If you send in a solution please use mathematics that a Stage 4 reader can follow.

If this problem caught your interest and you know some Stage 5 mathematics this Plus article on Carbon Dating could be a good next step for you.