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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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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?

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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

This activity could just be a warm-up, to draw a class into some mathematical thinking at the start of the lesson. Or it could be the first example of a more open investigation based on place value.

If I did want to make this a more extended activity I might begin with the "X marks the Spot" puzzle as an example, let the class together try it and discuss it, then invite children to create a puzzle like this for themselves, but to start with something simpler: Multiply two numbers, hide a few digits, then swap with a friend. Try each other's puzzle, check and discuss together, then share puzzles with other pairs.

There's lots to discuss, of course, but I would want to include consideration of puzzles that have more than one solution.

If none arise naturally from the class, I'd ask them to find the most digits that can be removed until a solution is no longer unique.

Differentiation by challenge is easy enough to achieve. I can control the size of the numbers that children use, and also vary the balance between the use of reasoning and trial and error.

Notes on the construction of X marks the Spot :

For help with Spinners, look back at October 02 . Increment buttons are very easy to insert and keep attention on the changing numbers, instead of the keyboard.

I have used Conditional Formatting to create a colour change ( to bright red ) when the right answer occurs. And have also used Conditional Formatting to conceal the ten-millions digit if it's zero ( font colour matched to background ) , and also to turn the border on or off for that cell.

Cell K6 does the multiplication, before the digit boxes each receive their correct value from the digits in that result. Although column & row headers have been turned off in Tools/Options/View, the location reference for the current cell can still be seen beside the Formula bar, immediately above the worksheet.

The individual digits are isolated using the INT function. INT takes just the integer part of a value and ignores the rest, for example INT of 3.845 is just 3. Using G2 as an example, this cell calculates the hundreds digit with the formula : = INT ( $K$6 / 100 ) - 10 * INT ( $K$6 / 1000 ) K6 was only made an absolute reference, $K$6 , to help me copy a basic formula across all the cells for digits and then just adjust the number of zeros required in each formula. INT ( K6 / 100 ) calculates the number of whole hundreds in the multiplication product. And from this I need to subtract 10 times the number of whole thousands in the product.

This technique of isolating digits was also used in the Happy Numbers investigation.