Several people sent in the correct answer to this problem. Phoebe and Lea, both 11 years old at Cottesmore School tell us how they worked through it:

We found this out by adding 6 and 5 together which makes 11.
We wrote out squared numbers to 100 and then we realised that the only squared numbers with a difference of 11 between them was 25 and 36. Therefore the numbers had to add up to 36.
We then did 20 + 5 and then 11 which all makes up to 36.

Lucy and Melissa who are in Year 5 at Woodfall Junior School explain which number is on each child's back:

Bob's number is 11, Mona's 20 and Jamie's 5.
Mona saw 11 + 5 = 16
Bob saw 20 + 5 = 25.
Jamie looked at Bob and Mona and got 31 which is 5 less than the square number 36 and 6 more than 25.