### 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.

### Patio

A square patio was tiled with square tiles all the same size. Some of the tiles were removed from the middle of the patio in order to make a square flower bed, but the number of the remaining tiles was still a square number. What were the dimensions of the patio and the flower bed?

### 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

### 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).

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.