Sums of Squares in Rectangles: Middle Secondary Mathematics Competition Question

In this question, we’re going to combine the sums of squares with a geometric interpretation.

Firstly, let’s look at how to write a number as sums of squares. This is called a number partition into squares. Let’s look at the number 12 and start with the most obvious partition into sums of unit squares.

12 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

It is important to do this in some kind of order, so let’s see how many partitions we can write using the next highest square, 22.

12 = 2² + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
12 = 2² + 2² + 1 + 1 + 1 + 1
12 = 2² + 2² + 2²

We have one more square that we can use and that’s 3².

12 = 3² + 1 + 1 + 1

That’s it! There are just 5 different ways to partition the number 12 into sums of squares. The next step is to look at the geometry of these partitions. Below are just three of these partitions. We can see that all the squares fit together to make rectangles with areas also equal to 12. However, the layout with the least number of squares has just 3 shapes. This is what we shall be looking for: the sum of squares with the least number of squares and that they fit within a rectangle with the same area as our original number.

Let’s see what happens to the number 11. It has similar partitions to 12 but with one unit removed.

11 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
11 = 2² + 1 + 1 + 1 + 1 + 1 + 1 + 1
11 = 2² + 2² + 1 + 1 + 1
11 = 3² + 1 + 1

Now, if we try to create rectangles with the sums of squares, we find that there are gaps. Imagine those rectangles for the number 12 with one unit square removed. However, this means that the smallest rectangles are all 12 units in area, apart from one. The only rectangle that has the same area as the number itself is the sum of unit squares.

So, let me recap. We are going to partition some numbers into their sums of square numbers. Then we are going to fit together the actual squares to create rectangles that have the same area as the original number. If there is more than one such rectangle we shall select the one with the smallest number of squares.

Try this with the numbers 30 to 40 inclusive. What do you notice about the prime numbers? What do you notice about the square numbers?




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