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# Weights

##### Age 11 to 14 Challenge Level:

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:

First of all we tried to find the solution by dividing all of the numbers between us and finding ways to reach these numbers. e.g.
-20 = D+C+B+A
-21 = D+C+B
-22 = D+C+2B+2A
-23 = D+C+2B+A

Using this method we couldn't find any solutions that didn't occur. However, we noticed a pattern in the weights, that each weight was -3 times the previous weight. In effect this means that the solutions are the numbers written in base -3.
This means that the first weight (A) are 1s, (B) are -3s, etc.
Since for each weight we can have 3 possible values (0, 1 or 2 weights), the number required for a -3 based system, we can make any of the numbers in the range.

In this base system any number can be written in only one way - just as in base 10.

Using this theory we quickly worked out the best values for the extension.
Since for each weight there are 4 possible values (None, 1, 2 or 3 weights) this will be a base -4 system, so the weights must be (since the most basic unit 1 is required) 1, -4 and 16.
This could also be done with a positive 4, giving the same range but no negative values.

Thank you and well done to you all!