Oliver got frustrated sometimes with his somewhat limited set of red, blue, and green building bricks. Part way through a project, he often ran
out of bricks of one particular colour. So he invented some 3D printing
machines that transform a brick of one colour into a combination of
bricks of one or more colours. Each machine can also operate in reverse.
The red machine (R) converts 1 red brick (r) into 1 blue brick (b) and 1
green brick (g). This process and its reverse are represented by
R
R-
r bg and bg+r.
Similarly, for the blue machine (B) we have
B
B-1
and rgb.
.
brg
And for the green machine (G) we have
8 Grb and rbg.
The machines can be used on collections of bricks, performing one con-
version at a time. For example, 3 blue bricks can be converted into 1
red and 3 green bricks in three steps, as follows:
B
B
bbb
rgbbrgrgb
rggg
Note that the order of the bricks in each collection is irrelevant.
a Show how Oliver can convert 1 red brick into 3 red bricks.
b Show how Oliver can convert 1 red brick into 5 red bricks in six steps.
c Explain why Oliver cannot convert 1 red brick into 5 red bricks in less
than six steps.
d Oliver starts with a collection of bricks in which the combined number
of blue and green bricks is odd. Explain why he cannot end up with
a combined number of blue and green bricks that is even.