Thanks to all of you who sent in a solution. We can't show them all here, but here are a few examples of different ways that you approached the problem.
Lizzie from Belmont Primary noticed that the sequences of numbers increased by the same number each time, and used this to work out her answers.
Roberta explains her thinking very concisely:
You times the two numbers together to find the number of blue circles and then you take away one from each of the numbers and times them together to find the number of red circles.
Andrew, Shatin College, Hong Kong explains his solution with more examples:
In this puzzle, 2 by 2 results in 5 and 4 by 4 would result in 25 (16 + 9).
You can work this out by multiplying the two numbers together and then you get part of the answer. The other half can be worked out by taking 1 from each of the two numbers and then multiplying them together. Then add the two numbers and you get the answer.
2 by 10 would have 29 (2 times 10 add 1 times 9), 3 by 10 would be 48 since 3 times 10 add 2 times 9, 10 by 10 would be 181 (10 squared add 9 squared).
If you use this formula you can work out anything. For Example: 100 by 100. 100 squared is 10000. then 90 squared is 8100 add it together and you get the answer 18100.
Quite a few of you used algebra to explain your thinking. Here is the solution from Michael of St Joseph's RC Primary School: