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Susie sent in a table to show her solution to this problem:



Numbers that can be squares Numbers that can be sticks Numbers that can be rectangles Numbers that can only be sticks
1 2 6 2
4 3 8 3
9 4 10 5
16 5 12 7
  6 14 11
  7 15 13
  8 16 17
  9 18 19
  10 20  
  11    
  12    
  13    
  14    
  15    
  16    
  17    
  18    
  19    
  20    

 

Susie also thought:


The numbers that can be only sticks are the prime numbers ie $2, 3, 5, 7, 11, 13, 17$ and $19$.
 
Georgina from Greetland School agreed with Susie's table.  Georgina says she had a go at this problem when she wasn't very well on the last day so had to go home.  Well done!

 

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.