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 from Hotwells Primary 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).
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.