Several people sent in the correct answer to this problem. Phoebe
and Lea, both 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 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.

Kevin also sent in his work on this question:

We know that Mona's number plus Bob's number is 5 less than a
square and 6 more than a square. So these squares must be 11 apart.
I wrote out the first few squares, and saw that they get further
and further apart, and the only ones that are 11 apart are 25 and
36. So Mona's number plus Bob's number is 31. We know that when you
add them all up you get a square, so Jamie's number plus 31 is a
square. From the hint, all the numbers are less than 40, so Jamie's
number plus 31 is 36. So Jamie's number is 5. Then I found that the
only way we could make the rest of the problem work is to have
Mona's number as 11 and Bob's as 20 (or the other way round).