c) Explain why Oliver cannot convert 1 red brick into 5 red bricks in less than 6 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 fo blue and green bricks that is even.
c) Explain why Oliver cannot convert 1 red brick into 5 red bricks in less than 6 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 fo blue and green bricks that is even.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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c) Explain why Oliver cannot convert 1 red brick into 5 red bricks in less than 6 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 fo blue and green bricks that is even.
Transcribed Image Text:out of bricks of one particular colour. So he invented some 3D printing
Oliver got frustrated sometimes with his somewhat limited set, of se a
blue, and green building bricks. Part way through a project, he often Ta
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
→ bg
and bg
R-1
r.
Similarly, for the blue machine (B) we have
B
B-1
b
→ rg
and rg
→b.
And for the green machine (G) we have
G
→ 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:
B
bbb
G-1
→ rggg
B
→ rgbb º→ rgrgb
Note that the order of the bricks in each collection is irrelevant.
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