Within the framework of the Square of Nine, a distinctive pattern emerges as we examine the halfway points between certain square numbers.

When we look at even square numbers and the next higher odd square numbers, we notice something neat. These halfway points make a sort of slope that goes down toward the lower right corner of the grid.

For example,

- Think about the number 64 and the number 81. The halfway point between them is 72.5.
- Similarly, if we find the halfway points between 100 and 121, we get 110.5,
- And between 144 and 169, it’s 156.5.

You can see this interesting trend in the figure below, where a line shows where these halfway points are on the grid, sloping down and to the right.

Now, when we check out odd square numbers and the higher even square numbers nearby, there’s another cool alignment happening.

This time, the halfway points make a kind of slope that goes up toward the top left corner of the grid.

In the above figure, you’ll notice another line that shows where these upward-sloping halfway points are. This line helps us see the pattern of how these points go up and to the left on the Square of Nine grid.

- Let’s take the halfway point between 81 and 100 – it’s 90.5.
- We can also find the halfway point between 121 and 144, which is 132.5,
- And between 169 and 196, it’s 182.5.

Now, to wrap things up, let’s bring together both of these properties and connect the lines in a single chart.