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.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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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.
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-1
→ r.
and bg
R
r
bg
Similarly, for the blue machine (B) we have
B
b
→ rg
and
rg
→ b.
And for the green machine (G) we have
G-1
→ rb
and
rb
→ g.
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:
G-1
→ rggg
B
B
bbb > rgbb P→ rgrgb
Transcribed Image Text: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-1 → r. and bg R r bg Similarly, for the blue machine (B) we have B b → rg and rg → b. And for the green machine (G) we have G-1 → rb and rb → g. 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: G-1 → rggg B B bbb > rgbb P→ rgrgb
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