Susie sent in a table to show her solution to this problem:
Susie also thought:
Children from Rampart School in the US sent a very full solution. They said:
The prime numbers from $1$ to $20$ ($2, 3, 5, 7, 11, 13, 17$, and $19$) can only be sticks. Each prime number has only two factors, $1$ and itself, so none of them can make rectangles. They can only make sticks of dimension $1 \times $ the prime number.
The numbers that are neither prime nor square ($6, 8, 10, 12, 14, 15, 18, 20$) can make rectangles because they all have factors other than $1$ and themselves. For example, $20$ has the factors $1$ and $20$, $2$ and $10$, and $4$ and $5$.
This leaves the square numbers that, subsequently, are the only numbers that can form the squares, for obvious reasons. Every square number can have the form $n \times n$, which also relates to the dimensions of the square.
So, we notice that only square numbers can form squares; prime numbers form sticks; and the composite, non-square numbers form rectangles.
Thank you also to Jack from Allerton Grange, Sophie from Belgium and Nathan from Rushmore Primary who sent in well-explained solutions.