Weights

We received a complete set of solutions, showing that all the numbers between -60 and 20 could be done, from Fred, Chester and Tom of Hotwells Primary School, by Jordan from Isleworth and Syon School and from Josh B, Matthew H, Seb W, Ben S, Max G and Jamie W, all in 8P1 at Hove Park School. Here is the solution from the Hove Park students (addition signs have been omitted so it is easier to see the pattern that forms).

When you are faced with such a problem it is helpful to work systematically. An anonymous solver used a stem leaf diagram.

It is possible to make every number between -60 and 20. I found this out by doing a series of different
combinations of the weights. I started by adding weight A to weight B then 2A to B then 2B to A and then 2A to 2B. After I found out every possible combination, I was left with 81 answers. I wrote all
the numbers out on a number line and used a stem leaf diagram to put them in size order.

The stem leaf diagram made it easy to see if there were any missing numbers, which there were not.
Here is the stem leaf diagram that I drew out.

 2|0
 1|0,1,2,3,4,5,6,7,8,9
 0|0,1,2,3,4,5,6,7,8,9,-1,-2,-3,-4,-5,-6,-7,-8,-9
-1|0,1,2,3,4,5,6,7,8,9
-2|0,1,2,3,4,5,6,7,8,9
-3|0,1,2,3,4,5,6,7,8,9
-4|0,1,2,3,4,5,6,7,8,9
-5|0,1,2,3,4,5,6,7,8,9
-6|0

For the first part of the problem, we also received this great solution from Rhea in Loughborough High School. She found a convincing argument for why all the numbers between -60 and 20 can be made, which didn't require her to write down a combination for every single number between -60 and 20.

The Maths Challenge Group at Colyton Grammar School explained how they approached the problem: