We had lots of responses to this problem - thank you to you all. Lachlan from St Faith's School sent a solution to the first part of the question:

two sevens =14
four fours=16

Ann and Daniel who are both 10, wrote this a slightly different way:

7+7=14
4+4+4+4=16

Ann and Daniel then continued with the rest of the question, writing the sums in order of size. Julia from St Anne's C of E Primary School and Michael from St Alfred's College also tackled the problem in this way. Here is Michael's solution:

4=4
7=7
4+4=8
4+7=11
4+4+4=12
7+7=14
4+7+4=15
4+4+4+4=16
4+7+7=18
4+4+4+7=19
4+4+4+4+4=20
7+7+7=21
4+7+4+7=22
4+4+4+4+7=23
4+4+4+4+4+4=24
4+7+7+7=25
4+7+4+7+4=26
4+4+4+4+4+7=27
7+7+7+7=28
4+7+4+7+7=29

Dylan, Robert, Shani and Amy who are all 7 years old, approached it a slightly different way (and have come up with some different solutions) which is just as good, although they have missed out a few:

4=4
4+4=8
4+4+4=12
4+4+4+4=16
4+4+4+4+4=20
4+4+4+4+4+4=24
4+4+4+4+4+4+4=28

7=7
7+7=14
7+7+7=21
7+7+7+7=28

7+4=11
7+4+4= 15
7+4+4+4=19

7+7+4=18
7+7+4+4+4=26
7+7+4+4+4+4= 30

Can you see where the missing sums would fit into the pattern they've created?

In conclusion, Michael says that the impossible numbers are: 1,2,3,5,6,9,10,13,17.