We received solutions to this problem from Nisarg, Kim and Grace. (Grace goes to Stradbroke Primary School.) Kim said:I knew I could tessellate the first box because the number of squares in it were divisible by three. I divided by three because the triomino was made of three boxes.
Here is a picture of the solution (you can still see how it is made, even though the "well done" message covers some of it up!):
Nisarg wrote a similar explanation about why you could fit the trionimoes into the first box but not the second.
Is it only about the number of squares in the box being a multiple of $3$?
Could you fit the trionimoes into the box below?