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Well done to Edison, from STC School in Hong Kong, and to Esther, who both sent us correct solutions. Here is Esther's solution:

The first thing I did was to list all the combinations of + and - signs that were possible and then I tried them out on two sets of consecutive negative numbers.

I noticed that the answers to some of them were the same so I tried to work out the general formulae using consecutive numbers $n$, $n+1$, $n+2$, $n+3$.

The results I got are in the table below:

Operations -8, -7, -6, -5 -10, -9, -8, -7 Formula
+ + + -26 -34 4n + 6
+ + - -16 -20 2n
+ - + -14 -18 2n + 2
+ - - -4 -4 -4
- + + -12 -16 2n + 4
- + - -2 -2 -2
- - + 0 0 0
- - - 10 14 -2n - 6