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Robert from Matthew Pearce Public School sent a solution to the first part of this problem. Robert says:

put $3, 9, 5, 7, 1$ in odd numbers and less than tenput $2, 4, 6, 8$ in even numbers and less than ten

put $17, 11, 13, 15, 19$ in odd numbers and not less than ten

put $10, 12, 14, 16, 18$ in even numbers and not less then ten

So, the completed Carroll diagram would look like this:

Well done also to Eliza from Danebank School who also completed this diagram.

Rowena from Christ Church Primary told us how she went about filling in the second Carroll diagram:

I decided the numbers that were between $10$ and $20$ did not
include $10$ or $20$, this left $11-19$, all these numbers have an
odd tens digit so they went in the bottom left corner. All the
remaining numbers must be in the 'Not between $10$ and $20$'
column, $10$ and $30$ have odd tens digit so they went at the
bottom of that column, all the $20$'s went in the top. I decided
that $0$ is an even number, so $1-9$ have even tens digit.

Here is a picture of Rowena's completed diagram:

You've explained this very well, Rowena. It's interesting that you've decided that the numbers $1-9$ have zero as their tens digit and that zero is even. Perhaps some of you have comments about this?