148
Solutions to Exercises
A
B
C
Initial position
1,2,3
*
*
Move 1
2,3
1
*
Move 2
2,3
*
1
Move 3
3
2
1
Move 4
3
1,2
*
MoveS
1,3
2
*
Thirteen moves are required.
By reversing the
Move 6
1,3
*
2
steps, we see that thirteen moves also suffice when
Move 7
3
1
2
the tower is initially on the middle peg.
Move 8
3
*
1,2
Move 9
*
3
1,2
Move 10
*
1,3
2
Move 11
1
3
2
Move 12
1
2,3
*
Move 13
*
1,2,3
*
(b) To transfer
n
discs from peg
A
to peg
B,
do the following:
1.
Transfer the top
n
 1 discs to peg
B
in
anl
moves.
2. Then transfer all
n
 1 discs from peg
B
to peg C in another
anl
moves.
3. Transfer the largest disc from
A
to
B
in one move.
4. Finally transfer the
n
 1 discs from peg C to peg
B
in another
anl
moves.
We obtain
an
=
3an
l
+
1.
(c) We obtain easily that
Pn
=
! is a particular solution to the recurrence relation
an
=
3an
l
+
1.
The characteristic polynomial associated with
an
=
3an
l
is
X2

3x
with characteristic roots
0,3. Thus,
qn
=
c(3
n
)
and
Pn
+
qn
=
c(3
n
)

!. Since
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 Summer '10
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 Graph Theory, Characteristic polynomial, 4 m, Recurrence relation, 3 L, 2 1,523,360 seconds, 3 00,000 centuries

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