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#56 Square sums by four

Set by Katie Steckles

1² = 1, 1² + 2² = 5 and 1² + 2² + 3² = 14. If we continue in this way, when is the first time this sum is a multiple of 4?

If we added the squares of all the numbers from 1 to 20, what would the remainder be after dividing by 4?

What is the general pattern in the remainders for the sum of squares of the numbers from 1 to n? Can you explain why this occurs?

Solution next week

#55 Squares in squares

Solution

As well as the original 9 by 9 frame holding a 3 by 3 photo, a set of 72 tiles can be used to make an 11 by 11 frame that holds a 7 by 7 photo, as 112 – 72 = 121 – 49 = 72. A 19 by 19 frame that holds a 17 by 17 photo can also be made, because 192 – 172 = 72. For square frames with a photo in the centre, the number of tiles needed will always be a multiple of 4. This can be shown by taking the frame size as a and the photo size as b, so the number of tiles is a2 – b2 or (a – b)(a + b). Since the photo is in the centre, a and b must either be both odd or both even, so (a-b) and (a+b) are both even and their product is a multiple of 4. To determine the possible frames for any number of tiles, divide the number of tiles by 4 and find all the rectangles with that area.