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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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Time of Birth

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

Excel Interactive Resource: X Marks the Spot

Challenge Level: Challenge Level:1

First the puzzle:

When a five-digit number, with "1" as the thousands digit, is multiplied by 4 1 7 the result is a seven-digit number which begins with 9 and ends 0 5 7. Find the missing digits for both these numbers.

Here's the Excel file: X marks the Spot.xls (Right-click on the link, "Save Target As", and select where you want the file to be saved).

X marks the spot
The increment buttons (called spinners in Excel) change the values of the digits, and automatically produce new results for the multiplication.

The answer to the puzzle is 2 1 9 2 1 but a class are very unlikely to get that just by flicking though digit values.

Many of us tackle this kind of puzzle with a mixture of reasoning and trial and error (or trial and improvement):
  • The units digit has to be 1 for the result to end in 7
  • The first digit has to be 2 to get 9 million and something.
Even when we hit upon a solution by trial and error, it's good to ask whether we can now see a reason why that had to be the answer.