161265

First, we noticed that if we align the rods from shortest to longest, each rod is 1 unit longer than the rod that precedes it. This made it easy to see that the squares are made from two color rods that are next to each other on this alignment.
Each figure has a boundary that is made of the bigger rod (we can also think of it as an unfilled square) and the inner filled portion is a square made of the preceding smaller rod. Together, the boundary and the inner square make a two-colored square.
If the boundary is made of pink rods, the inner square is made of the light green rod because it precedes it.
So, the next square in this series will be a square with a yellow boundary and inner square filled with pink rods. The next figure will have a dark green boundary and inner square of yellow rods.
We noticed a pattern that is shown in the attached file. Based on the data in the table, if n denotes the rod number on the alignment, the inner square makes n X n array and the two-colored square is (n+2) X (n+2) array. The exception is the white rods because they fit together with no inner square.
We noticed that the boundary rods are arranged like a jigsaw puzzle such that they do not overlap. We wondered if we made the boundary by creating overlaps, we could make squares that look different. Please see attached file. The squares were smaller but interestingly, we found that (if n denotes rod number on the alignment), the inner square makes n X n array and the two-colored square is
(n+2) X (n+2) array. This was the same as the previous case. In the previous case, there is no inner square when white rods are used but in this case, there is no inner square when white or red rods are used.